Comportement mécanique des matériaux des matériaux ... · Comportement mécanique des matériaux...
Transcript of Comportement mécanique des matériaux des matériaux ... · Comportement mécanique des matériaux...
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
ProblemProblem of of InterestInterest: : PolycrystalPolycrystal BehaviorBehavior
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 !
ProblemProblem of of InterestInterest: : PolycrystalPolycrystal BehaviorBehavior
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
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, ..)
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
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, ..)
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
FullFull--Field Computation: Effect of Local AnisotropyField Computation: Effect of Local Anisotropy
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
FullFull--Field Computation: Effect of Local AnisotropyField Computation: Effect of Local Anisotropy
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.