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  • Olivier CASTELNAU

    Comportement mcanique Comportement mcanique

    des matriaux des matriaux polycristallinspolycristallins

    Ecole MECANO, Autrans, 14-19 mars 2010

  • ProblemProblem of of InterestInterest: : PolycrystalPolycrystal BehaviorBehavior

    Lattice Preferred Orientationm

    orp

    ho

    log

    y

    )(),( xx

    + possibly other mechanical phases: grains boundaries,

    Gra

    in m

    orp

    ho

    log

    y

    Loading

    Solve for the mechanical problem at time t :

    + Microstructure evolution at large strain

    0 =div derives from u

    )(g =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)(),( xx

    ,

    Lattice Defects

    ScaleScale TransitionsTransitions

    km mm m nm

  • PlanPlan

    1- Microstructures- Textures morphologiques- Textures cristallographiques

    2- Mise en vidence des htrognits de champs, et implications- Champs cinmatiques (dplacements / dformations)- Champ statique (contraintes)

    3- Modlisation- 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 m

    O2 diffusion

    Zr tube (DESIROX test,JL Bchade)

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

    Zr02

    (O)Prior-

    30 nm

    Thin films (PHYMAT Poitiers)

  • IF steel, cold rolling (PhD. A. Wauthier, 2005 2008, H. Rgl + 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

    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 vidence des htrognits de champs, et implications- Champs cinmatiques (dplacements / dformations)- Champ statique (contraintes)

    3- Modlisation- 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 CorrelationDisplacement / Strain fields from Digital Image Correlationinitial deformed

    Michel Bornert (UR Navier / LMS-X), Jrome Crpin (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 =S

    C-axis

    Basal plane

    5.045

    090ou0

    ====

    S

    S

    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, Crpin, & 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

    )(:)()( . xxCx l=

    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 (mono- / poly-crystal)

    K sin21

    hkld

    )(KI

    2

    lIncident beam

    (mono- / poly-chromatic)

    Kn //hkl

    Shift of diffraction lines :

    )(:)(2

    xKK

    x lastKKK

    =

    = ddddfI KK ...),,,()( 0 KkK

    Spatial resolution

    Instrumental response

    SCALAR !

    Field measurement in the 2-D orientation space

    [ ] >

  • Grain size ~ 50m (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. Bchade, CEA; T. Chauveau, B. Bacroix, LPMTM)

    Zr, 15% plastic strain

    prismatic dislocs

    intragranular stress (res) fluctuations ~100MPa

  • Interpretation of Line ShiftsInterpretation of Line Shifts

    - Elastic behavior - Thermal

    elastic response

    xBx :)()( =

    inelastic response

    )(xres

    +0 =

    TWO CONTRIBUTIONS !!

    B: localization tensor

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

    +

    =>< res22 ::::: SKK

    BSKK

    KKK

    general expressionfor " " law2sin

    )(:)( xSx =

    sometimes leading term,often omitted

  • Validity of the "sinValidity of the "sin22 law"law"

    K

    1

    2

    3

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

    +=>< res22 ::::: S

    KKBS

    KK

    KKK

    )sin;sincos;cos(cos =K1

    ( )( )

    ( ) 2+++

    ++++

    +++=>=

    10

    -3

    -0.15%

    -0.10%

    -0.05%

    0.0 0.2 0.4 0.6 0.8sin2

    X-R

    ay S

    trai

    n (%

    )

    T7

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