Comportement mécanique des matériaux des matériaux ... · Comportement mécanique des matériaux des matériaux polycristallins polycristallins Ecole MECANO, Autrans, 14-19 mars

Olivier CASTELNAU

Comportement mécanique Comportement mécanique

des matériaux des matériaux polycristallinspolycristallins

Ecole MECANO, Autrans, 14-19 mars 2010

ProblemProblem of of InterestInterest: : PolycrystalPolycrystal BehaviorBehavior

Lattice Preferred Orientationm

orph

olog

y

)(),( xεxσ

+ possibly other mechanical phases : grains boundaries, …

Gra

in m

orph

olog

y

Loading

σ

Solve for the mechanical problem at time t :

+ Microstructure evolution at large strain

0σ =divε derives from u

)(σgε =Local constitutive relation

Stress equilibrium

Boundary conditions

scal

e


Young modulus vs. crystallographic direction

Case 1 (simple) : isotropic local behavior + single phase

crystallographic direction

Homogeneous material (from the mechanical point of view)The behavior is microstructure independent !


Young modulus vs. crystallographic direction

Case 2 (usual) : ANisotropic local behavior (+ many phases)

crystallographic direction(case of Au )

HETEROgeneous material (from the mechanical point of view)The behavior is microstructure DEpendent !

ExampleExample: : IceIce RheologyRheology

IngredientsIngredients

Geometrical arrangement of constituents(volume fraction, shape, spatial arrangement, crystall. orientation, …)

Mechanical response of constituents

Experimental data at different scalesExperimental data at different scales

Micromechanical model (bridges constituent and overall scales)

Large Scale Flow

Representative Volume Element

Collective Behavior of

Lattice Defects)(),( xεxσ

εσ,

Lattice Defects

ScaleScale TransitionsTransitions

km mm µm nm

PlanPlan

1- Microstructures- Textures morphologiques- Textures cristallographiques

2- Mise en évidence des hétérogénéités de champs, et implications- Champs cinématiques (déplacements / déformations)- Champ statique (contraintes)

3- Modélisation- En champs complets (full-field) par FFT- En champs moyens (bornes Reuss, Voigt, estimations VW, SC, ..)

MorphologicalMorphological TTexturesexturesSpatial arrangement of grains (TEM, SEM, Tomography, EBSD, …)

EBSD (Electron Backscaterring Diffraction = Orientation Imaging Microscopy

Polycrystal MicrostructuresPolycrystal MicrostructuresEBSD analysis of ETP Copper

- Large grain size distribution- Complex and various grain shapes- Microstructure randomness

2 possibilities :

A small number of large and nice grains3-D characterization may be possible (eg. 3DRXD)

Otherwisetry to find out a statistical representation of the microstructure or

a model for the microstructure

A simple random model : Voronoi tesselation

Microstructure Microstructure ModellingModelling

etc…

Microstructure Microstructure ModellingModelling

Voronoi tesselationCu from EBSD,

filled with 8 orientations

8 mechanical phases (crystal orientations / colors),

8500 grains

More More PolycrystalsPolycrystals MicrostructuresMicrostructures

600 µm

O2 diffusion

Zr tube (DESIROX test,JL Béchade)

Fe-Ni + Olivine meteorite(L.A. Science Center)

Zr02

α(O)Prior-β

30 nm

Thin films (PHYMAT Poitiers)

IF steel, cold rolling (PhD. A. Wauthier, 2005 – 2008, H. Réglé + B. Bacroix, ARCELOR)

EBSD:fragmentationmisorientation

Microstructure Evolution (plastic strain)Microstructure Evolution (plastic strain)

15% 40%15% 40%

undeformed deformed (~1%)Ice (M. Montagnat et al., LGGE)

CrystallographicCrystallographic TTexturesextures

The Orientation Distribution Function (ODF)

ggg

dfV

dV)(

)( =

density probability of grains with crystallographic orientation g

a function in the 3-D Euler space

Hot rolled IF steel

CrystallographicCrystallographic Textures : Textures : MMeasurementseasurements

X-ray / neutrons diffraction

(2-D) Pole figures

for several hkl planes

θλ sin2 hkld=Bragg law :

(3-D) ODF calculation

Each pole figure is an integration of the ODF

Different methods (spherical harmonics, vector decomposition, …)

stereographic projection

CrystallographicCrystallographic Textures : Textures : EffectsEffectsZr 702 specimens, channel die compression

Local anisotropy + crystallographic texture macroscopic anisotropy

CrystallographicCrystallographic Textures Evolution Textures Evolution atat large large strainstrain

Like packs of cards…

hardsoft

Zr 702 specimens, biaxial deformation

initial state after strain of ~ 0.45

PlanPlan




Ex.: Plasticity of Zirconium AlloysEx.: Plasticity of Zirconium Alloys

grain

15% plastic strain

boundary

traces of dislocationsslip systems

gold microgrid(initially square)

• Huge intra- and inter-granular strain heterogeneity• Heterogeneous activation of deformation mecanisms

deformation twins

Displacement / Strain fields from Digital Image Cor relationDisplacement / Strain fields from Digital Image Cor relationinitial deformed

Michel Bornert (UR Navier / LMS-X), Jérome Crépin (CdM), GdR2519,…

IntragranularIntragranular StrainStrain HeterogeneityHeterogeneity in in IceIce(PhD Fanny Grennerat, LGGE, M. Montagnat, ...)

Columnar Ice« 2-D » specimen, in-plane c-axis

Grain orientation provided by a single parameter(Schmid factor)

)2sin(21 θ=S

C-axis

Basal plane

θ

5.045

090ou0

=⇒°==⇒°°=

S

S

θθ « soft »

« stiff »

Macroscopic strain : 1% Macroscopic strain : 2.3%

IntragranularIntragranular StrainStrain HeterogeneityHeterogeneity in in IceIce

Equivalent strain distribution (log scale)

• Deformation in localized• Strong intra- and inter-granular heterogeneities ! • Band extend ~ few grain size• Localization sharpness increases w/ strain

• Similar results for other (eg. metallic) materials(see work by Bornert, Crépin, & Co. at LMS-X)

Schmid factor distribution

soft

stiff

softstiff

Schmid factorLocal strain vs. Schmid factor

IntragranularIntragranular StrainStrain HeterogeneityHeterogeneity in in IceIce

Local strain vs. Schmid factor

Schmid factor

softstiff

Equ

ival

ent s

trai

n(n

orm

.)

• Grains w/ large S do not necessarily deforms rapidly !• Grains w/ small S can deform significantly !

Static (/ Stress) Field from Diffraction TechniquesStatic (/ Stress) Field from Diffraction Techniques

Accuracy (absolute) ~10-4

)(:)()( . xεxCxσ él=

Partly measured by X-Ray DiffractionWanted

Accuracy (absolute) ~10

6 independent components (+ 3 orientation angles)

not always (rarely) an easy task !

Diffraction : principlesDiffraction : principles

Spatial Instrumental

V

K

k0

kh

Diffraction vector

Diffracting volume ΩΩΩΩ(mono- / poly-crystal)

K λθsin21

≈hkld

)(KI

θ2

lIncident beam

(mono- / poly-chromatic)

Kn //hkl

Shift of diffraction lines :

)(:)(2

xεKK

x élastKKK

⊗=ε

∫∫∫ ∫ Ω= λδθε ddddfI KK ...),,,()( 0 KkK

Spatial resolution

Instrumental response

SCALAR !

Field measurement in the 2-D orientation space

[ ] Ω><−== ∫ KKKdssIssI εµ )(.)()1(

Au film(Faurie,LURE)

Shift of diffraction lines :

Grain size ~ 50µm (ID22, ESRF)

reference powder

diffraction line width

Grain #1

IntragranularIntragranular XRD: XRD: plasticity of plasticity of ZrZr alloysalloys

intragranular

(T. Ungar, G. Ribarik, Univ. Budapest; M. Drakopoulos, A. Snigirev, I. Snigireva, ESRF; B. Lengeler, C. Schroer, RWTH Aachen; J.L. Béchade, CEA; T. Chauveau, B. Bacroix, LPMTM)

Zr, 15% plastic strain

prismatic dislocs

intragranular stress (σσσσres) fluctuations ~100MPa

Interpretation of Line ShiftsInterpretation of Line Shifts

- Elastic behavior - Thermal

elastic responseσ

σxBxσ :)()( =

inelastic response

)(xσres

+0σ =

TWO CONTRIBUTIONS !!

B: localization tensor

- Elastic behavior - Thermal- Plasticity, viscoplasticity- Twinning, phase transition- …

ΩΩΩ⊗

+⊗

=>< res22::::: σS

KKσBS

KK

KKKε

general expressionfor " " lawψ2sin

)(:)( xσSxε =

sometimes leading term,often omitted

Validity of the "sinValidity of the "sin 22ψ ψ ψ ψ ψ ψ ψ ψ law"law"

K

ϕ

ψ

1

2

3σ

Assumptions : purely elastic response, local elasticity isotropic ( ), no residual stresses ( )0σ =resIB =

ΩΩΩ⊗+⊗=>< res22

::::: σSKK

σBSKK

KKKε

)sin;sincos;cos(cos ψϕψϕψ=K1

( )( )

( ) 2ψϕ+ϕ++

++−++

ψ−ϕ+ϕ+ϕ+=>=<

sinsincos1

1

sinsin2sincos1

2313

33221133

233

22212

211

σσν

σσσνσν

σσσσνεε Κϕψ

E

EE

EDΩ

TensileTensile Tests on Tests on ElasticElastic ThinThin Films Films (Goudeau, Renault, Faurie, Le Bourhis, & Co)

1,5 420 Experiments

x1

x2

1 1 1

Min =-1.23E+00 Max = 2.57E+01

LPMTM-CNRS

x1

x2

1 1 1

Min =-1.23E+00 Max = 2.57E+01

LPMTM-CNRS

x1

x2

1 1 1

Min =-1.23E+00 Max = 2.57E+01

LPMTM-CNRS

x1

x2

1 1 1

Min =-1.23E+00 Max = 2.57E+01

LPMTM-CNRS

111 fiber texture

Au film: local & macro anisotropy

0.00%

0.05%

0.10%

0.15%

0.20%

X-R

ay S

train

(%)

T0T1T2T3T4T5T6T7

W film: local (& macro) isotropy

0,0 0,2 0,4 0,6 0,8 1,0

-0,5

0,0

0,5

1,0

1,5 420

311222

420

331

420311

331222

ε (x

103 )

sin2 Ψ

Experiments r=0.1SC model

< ε K

K>

Ω×

10-3

-0.15%

-0.10%

-0.05%

0.0 0.2 0.4 0.6 0.8sin 2ψψψψ

X-R

ay S

train

(%)

T7

• Unique technique to measure elasticstrain at a local scale• Requires micromechanical modellingfor the interpretation

PlanPlan




Large Scale Flow

Representative Volume Element

Collective Behavior of

Lattice Defects)(),( xεxσ

εσ,

Lattice Defects

ScaleScale TransitionsTransitions

km mm µm nm

MicromechanicalMicromechanical ApproachesApproaches

Microstructure

Predictive model (micromechanics)

Local behavior0...),,,,( =σσεε &&f

EBSB, XRD, SEM, TEM, Tomo X, ..

monocrystal, DDD, TEM,

identification, …

Effective behaviorField distributions

Identification / Validation

mechanical tests, 2/3-D strain field,stress field, …

Micromechanical modellingMicromechanical modelling

LPOG

rain

mor

phol

ogy

Solve for the mechanical problem at time t :

)(),( xεxσ &

A complex problem !!! Two strategies :

+ Microstructure evolution at large strain

0σ =divε& derives from u

)(σgε =&Local constitutive relation

Stress equilibrium

Boundary conditions

cm

Gra

in m

orph

olog

y

Loading

A complex problem !!! Two strategies :

Solve it numerically (Finite Element, Fourier, ..)- "Exact" solution for the considered microstructure- Requires significant numerical efforts

Solve it theoretically (homogenization techniques)- Keep same LPO- Simplified description of the microstructure- Approximate solution (good enough ?)- FAST !

σ

cm

( ) ( )( ) ( ) ( )( ) ( )xτxεc

xεcxcxεc

xεxcxσ

+=

−+=

=

:

:)(:

:)(

0

00

Heterogeneous elastic material

EquilibriumContinuous displacement field (Green function method)

FFT Full Field ComputationFFT Full Field Computation[Moulinec-Suquet 1998, Lebensohn 2001]

)(),(),( xεxσxc

Homogeneous elastic material w/ polarisation

Continuous displacement fieldConstitutive behaviorBoundary conditions

(Green function method)

E0ε

0ξξτξΓξεxτxΓExε

=≠∀=⇒∗−=−

)(ˆ

,)(ˆ:)(ˆ)(ˆ)()()( oo FFT !

Fourier spaceDirect space

Advantages : (very) fast, low memory needsOK for infinite contrasts / non-linearities, incompressible composites

Limitations : periodic microstructure and boundary conditionsgrid refinement not possible

FullFull--Field Computation: Effect of Local AnisotropyField Computation: Effect of Local Anisotropy

unia

xial

ext

ensi

onOlivine crystal(Mg,Fe)2SiO4

viscoplastic behavior,

Microstructure generation: Voronoi tesselation

x

32 grains with random orientation643 Fourier points, i.e. ~8192 Fourier point / grain,

Periodic Boundary conditionsEnsemble average over 50 realizations (ergodicity)

unia

xial

ext

ensi

on

viscoplastic behavior,few slip systems,

with different yield stress

5.3,)()()(

0

1

00==

−

nn

ττ

ττ

γγ xxx&

&

eqeq σσ /)(x


Stress distribution

moderate local plastic anisotropy

(M=10)

high localplastic anisotropy

(M=100)

eqeq εε && /)(x

Strain-ratedistribution

Field distribution (phase)Field distribution (phase)

Field statistics in grains with a "soft" crystal orientation

Equivalent stress Equivalent strain rate

Broad distributions at large anisotropy M.

Local equivalent stress and strain rate globally larger than their macroscopic counterpart.

Field distribution (slip system)Field distribution (slip system)

Soft slip system in a soft crystal orientation

Resolved shear stress

Shear rate

Stress and strain rate distribution have different shapeLarge fluctuations at large anisotropy M

Bimodal distribution of shear stress

Reveals very strong intergranular interactions


each point =a different

position insidethe polycrystalthe polycrystal

macroscopicbehavior

n=1 n=3.33

FullFull--Field Computation: Effect of nonlinearityField Computation: Effect of nonlinearityStrain distribution in a 2D composite

Microstructure

(Moulinec, Suquet, Eur. J. Mech., 2003)

n=5 n=10

hard inclusionssoft matrix

neqeq σ

ε

ε

∝00

)()(

σxx

&

&

(contrast: 5)

isotropic behaviors

Full Field ComputationFull Field Computation

Very nice, very intructive,accurate solutions (reference),

BUTlenghty calculations (days / weeks / …)

not adapted for a coupling with large scale flow model

Linear homogenization: principlesLinear homogenization: principles

)(0

)(

0

)(:

)()(:)()(rr εxσm

xεxσxmxε

+=

+=)(

0)( , rrεm & homogeneous / phase

0~:~ εσmε +=

∑

∑

=>=<

=><=

rrrr

rrrr

c

c

)()(0

)(00

)()()(

::~

::~

BεBεε

BmBmm)(. r : phase average

Local behavior

Effective behaviorex.: thermo-elasticity

• B: localization tensor of the problem without eigenstrain• B: localization tensor of the problem without eigenstrain• Only the phase average B(r) is required → powerfull

)()()()()()()(

~2

:rr

rrres

rrr U

c mσσσσBσσ

∂∂=>⊗<+=>=<

First moment Second moment

Effective behavior + Field statistics (Inter- & Intra-phase fluctuations)

Relocalisation

[ ]><−= 0:::21~

εσσmσ resU

Simple SchemesSimple Schemes

ReussReuss (static) bound(static) bound

Uniform stress field within the polycrystal, B = IStrain homogeneity within grainsUpper bound for (i.e. )m~ σmσσmσ :~::~: R≤

Voigt (Taylor) boundVoigt (Taylor) bound

Uniform strain field within the polycrystalStress homogeneity within grainsLower bound for (i.e. )m~ σmσσmσ :~::~: V≥

An analytical solution for B for microstructures exhibiting perfect desorder [Kröner, 1978]

SelfSelf--Consistent (SC) Consistent (SC) schemescheme

RSCV mmm ~~~ ≤≤Accounts for intragranular and intergranular heterogeneities

FullFull--fieldfield vs. SC vs. SC estimatesestimates : : linearlinear casecase

Effective behavior Overall heterogeneities

NonlinearNonlinear homogenizationhomogenization

)(0

)( )(:)( rr εxσmxε +=

Find out the best linear constitutive relation with phase homogeneous modulus and stress-free strain

so that the nonlinear effective behavior is best predicted

NonlinearNonlinear homogenizationhomogenization

)(0

)( )(:)( rr εxσmxε +=

Second order method[Ponte-Castañeda, 2002][Ponte-Castañeda, 2002]

first moment first & second moments

Effective behaviorEffective behavior

1/2

= VPSC

[Masson et al, 2000]

[Molinari et al, 1987;

[Ponte-Castañeda, 2002]

- FFT : more than 3 independent slip systems are needed !

- SO estimate is excellent !

- Only SO provide the correct trend.

[Molinari et al, 1987;Lebensohn & Tomé, 1993]

Overall heterogeneitiesOverall heterogeneities

eqeqeq σσσ /22 −>< eqeqeq εεε &&& /22 −><

Only SO estimate predicts the correct trend

Field distributionsField distributions

eqeqr σ/][ )(>< σ

M = 100 SOFFT

eqeqr ε&& /][ )(>< ε

RésuméRésumé

• Lien microstructure – propriétés : comportement local (anisotropie), microstructure• Anisotropie locale ou non-linéarité du comportement → hétérogénéités des champs• Fortes interactions mécaniques entre les grains• Localisation des contraintes / déformation (bandes)• Attentions aux interprétations locales « à la main » : effet de la • Attentions aux interprétations locales « à la main » : effet de la microstructure locale potentiellement fort.• Complémentarité des modèles full-field / mean-field

Few Few referencesreferences

• S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag (2002)• G. W. Milton, The theory of composites, Cambridge University Press, 2001• H. Moulinec, P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, cmame, 157, p.69--94, 1998• M. Bornert, T. Bretheau, P. Gilormini, Homogénéisation en mécanique des matériaux(2 tomes), Hermès Science Publications, 2001.

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Comportement mécanique des matériaux des matériaux ... · Comportement mécanique des matériaux des matériaux polycristallins polycristallins Ecole MECANO, Autrans, 14-19 mars