[PPT] - 1 Micromechanics using Spectral Method Interface Decohesion in PowerPoint Presentation

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Micromechanics using Spectral Method

Interface Decohesion in Polycrystals

L. Sha

harma, P. Shanthraj, M.Diehl, F. Roters, D. Raabe,

R. Peerlings and M. Geers

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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

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Mo Motiv tivation ation

strain localization
damage initiation
recrystallization nucleation
…
tool design
crashworthiness
component properties
…

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Mo Motiv tivation ation

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http://www.virtualexplorer.com.au/special/meansvolume/contribs/jessell/labs/02a.m

v

Crystal Plasticity is always a multi-scale problem !

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Mo Motiv tivation ation

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simulation requirements

arbitrary mechanical

boundary value problems

continuum mechanics
accounting for crystal

plasticity Crystal Plasticity Finite Elemente Method (CPFEM) Or Crystal Plasticity Spectral Method (CPFFT)

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CP CPFEM/ FEM/CPFFT CPFFT st strat rategy egy

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M

material point model

𝐆 𝐐

solver for

equilibrium
compatibility

F P

deformation partitioning & homogenization crystallite elasto-plasticity

Fe

Lp S constitutive law

elasticity
plasticity

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Th The sp spectral tral so solv lver

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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

A little history

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Sp Spectral ctral me method hod

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div 𝝉 = 0 Static equilibrium: Split strain: 𝜻 = 𝜻 + 𝜻 𝜻𝑛+1 = 𝜻𝑛 − 𝜟 ∗ 𝝉𝑛

FFT

Material law: 𝝉 = 𝑫𝜻 Introduce reference medium: Stiffness 𝑫 𝜻𝑛+1 = 𝜻𝑛 − ℱ−1 𝜟: 𝝉𝑛

with Γ𝑗𝑘𝑙𝑚 = 𝑙𝑘𝑙𝑚 𝑂𝑗𝑙 and 𝑂𝑗𝑙 = 𝑙𝑚𝑙𝑘 𝐷𝑗𝑘𝑙𝑚

−1

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Co Comp mparion arion FEM FEM vs vs FFT FFT

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P. Eisenlohr., M. Diehl, R. A. Lebensohn, F. Roters: International Journal of Plasticity (2013), 37 - 53

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Ex Experim rimental ental-Numerical Numerical

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Example: Basal slip in Magnesium

F. Wang, S. Sandloebes, M. Diehl, L. Sharma, F. Roters, D. Raabe:Acta Materialia 80 (2014) 77-93

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Me Mesosc

scale

ale me mechan hanics ics

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High Resolution Crystal plasticity enabled through robust

spectral solvers Shanthraj et al. [IJP, 2015]

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

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Int nterface erface decoh

hesion

esion (form

rmab

ability ility li limi miter) er)

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Void Initiation Role e of th the e Int nter erfa face ces Sur urroun

undin

ding Microst

stru

ructur ture

?

Void Growth/Propagation Compu puta tati tion

nal to

tool to to model Inte terfa face ce decohesion ion

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Int nterface erface mo modeli ling ng of

f pol
lycrystals

crystals

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Interface elements Interface band

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Ei Eige gen n St Stra rain in Dama mage ge (Pandolf

lfi,

i, Ortiz z et al.; ; Menzel el, , Ekh et al., 2002)

)

Accomodation by eigen strain.
(interface-) plane stretching effects.
In an anisotropic way (normal and tangential modes).

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Field problem

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Dama mage ge re regulariz gularization ation so solv lved us usin ing g FFT FFT

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Utilize Fourier transform
Solved for its roots using

Jacobian free Newton method.

Hetrogenous regularization lengthscale

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Te Test st Si Simu mulatio lation

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Pol

lycryst

ycrystal al Si Simulation mulation

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Res

esolut ution

n: 256x256x2
Int

nter erfa face ce Band nd th thickne ness: 4 voxels

Ran

andomly ly orien enta tati tion

n

FCC

Elas

asto to-pl plas astic tic-dam damage age (cryst ystal plasti ticit ity) y)

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

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

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

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Fut Future ure wo work rk

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Coupling with damage models in the bulk
Monolithic schemes for Multiphysics
Implementation using petsc4py
Time integrators (Fortran support)

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Acknowl knowledgme edgment nt

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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

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Thank you. Questions?

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Si Simu mulat lation ion (hexag

xagon

nal

al polycry cryst stal al loaded ded hori rizo zont ntall ally )

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Si Simu mulat lation ion

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St Stra rain in lo localis lisation ation

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Br Britt ittle le Si Simu mulat lation ion

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Damage Strain (F11) Stress (P11)

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Me Mesosc

scale

ale me mechan hanics ics

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Crystal plasticity.
FFT based Spectral method

Shanthraj et al. [2015]

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Ex Experim rimental ental-Numerical Numerical

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Example: Basal slip in Magnesium

Wang et al. [2014]

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Ra Rate te in independe endent nt

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Ra Rate te in independe endent nt

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Local Damage

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1 for undamaged material
0 for fully damaged
Monotonously decreases

(irreversibility)

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Normal opening strains

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Tangential opening strains

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Stress Integration (the Local problem)

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Mesh Objectivity

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Coarse mesh (10 fourier points in the

band)

Fine = 2x coarse

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Wo Work rk of

f se

separation ration wi with h band nd thi hickness kness

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Vox

xelize

elized fie ield ld of

f the

he no norm rmals als

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Generator points of

standard voronoi tessellation.

First order cartesian

moments. [Libermann et al., 2015]

1 1 1 1 1 1 1 1 1 1 1 2 3 3 3 3 3 2 2 2 2 2 2 4 4 4 4