TD Elements Finis cristallins
Henry Proudhon, Georges Cailletaud, Samuel [email protected]
Centre des Materiaux, MINES ParisTech, UMR CNRS 7633
Ecole MECANO 15-19 Mars
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 1 / 44
Outline
1 Introduction
2 A crystal plasticity model
3 FE calculation using single crystal modelsPlaying with Schmid FactorSingle crystal notched sample
4 FE calculation using polycrystal models
5 Crystal plasticity computations on real microstructuresPolycrystalline β Ti under tensionSimulation of coherent diffraction in a polycrystalline film
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 2 / 44
Contents
1 Introduction
2 A crystal plasticity model
3 FE calculation using single crystal modelsPlaying with Schmid FactorSingle crystal notched sample
4 FE calculation using polycrystal models
5 Crystal plasticity computations on real microstructuresPolycrystalline β Ti under tensionSimulation of coherent diffraction in a polycrystalline film
Introduction
Evidences of crystallographic slip
Deformed pure polycrystalline copper (Nomarski contrast)
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 4 / 44
Introduction
Z-SeT/Zebulon
get the software: http://www.nwnumerics.com/Downloads/
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 5 / 44
Contents
1 Introduction
2 A crystal plasticity model
3 FE calculation using single crystal modelsPlaying with Schmid FactorSingle crystal notched sample
4 FE calculation using polycrystal models
5 Crystal plasticity computations on real microstructuresPolycrystalline β Ti under tensionSimulation of coherent diffraction in a polycrystalline film
A crystal plasticity model
Basic ingredients of crystal plasticity
normal
slipdirection
A collection of N slip systems
Normal to slip plane ns
Direction of slip system ls
Orientation tensor m∼s =
1
2(ns ⊗ ls + ls ⊗ ns)
Resolved shear stress τ s = σ∼ : m∼s
Yield function f s(τ s , hardening variables, disl. density,. . . ) = 0Elastic behavior ∀s, f s < 0Shear strain rate γs = γs(f s)
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 7 / 44
A crystal plasticity model
A single crystal model
Viscoplastic formulation, with additive hardening, to define a yieldfunction f s on each slip system s
Viscoplastic flow: Shear strain rate γs deduced from resolved shearstress τ s and from the value of the kinematic (X s) and isotropic (Rs)variables:
f s = |τ s−X s |−Rs v s =
⟨f s
K
⟩n
γs = v ssigne(τ s−x s) = v sηs
Hardening rules : X s and Rs computed by means of state variablesαs and r s
X s =cαs αs =(ηs − dαs)v s
Rs =r0 + Q∑
j
hsj rj r j =(1− br j)v j
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 8 / 44
A crystal plasticity model
Interaction matrix
B4 B2 B5 D4 D1 D6 A2 A6 A3 C5 C3 C1B4 h1 h2 h2 h4 h5 h5 h5 h6 h3 h5 h3 h6
B2 h1 h2 h5 h3 h6 h4 h5 h5 h5 h6 h3
B5 h1 h5 h6 h3 h5 h3 h6 h4 h5 h5
D4 h1 h2 h2 h6 h5 h3 h6 h3 h5
D1 h1 h2 h3 h5 h6 h5 h5 h4
D6 h1 h5 h4 h5 h3 h6 h5
A2 h1 h2 h2 h6 h5 h3
A6 h1 h2 h3 h5 h6
A3 h1 h5 h4 h5
C5 h1 h2 h2
C3 h1 h2
C1 h1
simplified version (Taylor model): h1 = h2 = h3 = h4 = h5 = h6 = 1
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 9 / 44
A crystal plasticity model
Small strain formalism
Strain partition:ε∼ = ε∼
e + ε∼p
Resolved shear stress, computed by means of the orientation tensor,m∼
s , using the vector normal to the slip plane, ns , and the slipdirection, ms
τ s = σ∼ : m∼s m∼
s =1
2(ns ⊗ms + ms ⊗ ns)
Viscoplastic strain rate (from the viscoplastic potential):
Ω(σ∼ , . . .) =∑
s
K
n + 1
⟨f s
K
⟩n+1
ε∼p =
∑s
∂Ω
∂σ∼=
∑s
∂Ω
∂f s
∂f s
∂σ∼=
∑s
v sm∼sηs =
∑s
γsm∼s
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 10 / 44
Contents
1 Introduction
2 A crystal plasticity model
3 FE calculation using single crystal modelsPlaying with Schmid FactorSingle crystal notched sample
4 FE calculation using polycrystal models
5 Crystal plasticity computations on real microstructuresPolycrystalline β Ti under tensionSimulation of coherent diffraction in a polycrystalline film
FE calculation using single crystal models Playing with Schmid Factor
Slip systems in FCC crystals
4 slip planes 111 i.e.(111), (111), (111), (111)
slip directions a 111–plane areindicated by arrows
number 1 2 3 4 5 6name B4 B2 B5 D4 D1 D6plane (111) (111) (111) (111) (111) (111)direction [101] [011] [110] [101] [011] [110]
number 7 8 9 10 11 12name A2 A6 A3 C5 C3 C1plane (111) (111) (111) (111) (111) (111)direction [011] [110] [101] [110] [101] [011]
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FE calculation using single crystal models Playing with Schmid Factor
Simple tension for a single crystal specimen
tensile test in the direction [001] 8 active slip systemsSchmid factor Ms = (ns .t) (ms .t)
number 1 2 3 4 5 6name B4 B2 B5 D4 D1 D6plane (111) (111) (111) (111) (111) (111)direction [101] [011] [110] [101] [011] [110]Schmid factor 1√
61√6
0 1√6
1√6
0
number 7 8 9 10 11 12name A2 A6 A3 C5 C3 C1plane (111) (111) (111) (111) (111) (111)direction [011] [110] [101] [110] [101] [011]Schmid factor 1√
60 1√
60 − 1√
6− 1√
6
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 14 / 44
FE calculation using single crystal models Playing with Schmid Factor
Simple tension for a single crystal specimen
tensile test in the direction [111] 6 active slip systemsSchmid factor Ms = (ns .t) (ms .t)
number 1 2 3 4 5 6name B4 B2 B5 D4 D1 D6plane (111) (111) (111) (111) (111) (111)direction [101] [011] [110] [101] [011] [110]Schmid factor 0 0 0 0 2
3√
62
3√
6
number 7 8 9 10 11 12name A2 A6 A3 C5 C3 C1plane (111) (111) (111) (111) (111) (111)direction [011] [110] [101] [110] [101] [011]Schmid factor 0 2
3√
62
3√
60 2
3√
62
3√
6
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 15 / 44
FE calculation using single crystal models Playing with Schmid Factor
Simple tension for a single crystal specimen
tensile test in the direction [011] 4 active slip systemsSchmid factor Ms = (ns .t) (ms .t)
number 1 2 3 4 5 6name B4 B2 B5 D4 D1 D6plane (111) (111) (111) (111) (111) (111)direction [101] [011] [110] [101] [011] [110]Schmid factor 1√
30 1√
30 0 0
number 7 8 9 10 11 12name A2 A6 A3 C5 C3 C1plane (111) (111) (111) (111) (111) (111)direction [011] [110] [101] [110] [101] [011]Schmid factor 0 1√
31√3
0 0 0
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 16 / 44
FE calculation using single crystal models Playing with Schmid Factor
Simple tension for a single crystal specimen
tensile test in the direction [012] 2 active slip systemsSchmid factor Ms = (ns .t) (ms .t)
number 1 2 3 4 5 6name B4 B2 B5 D4 D1 D6plane (111) (111) (111) (111) (111) (111)direction [101] [011] [110] [101] [011] [110]
Schmid factor√
65
15
√32
15
√32
15
√23
15
√32
15√
6
number 7 8 9 10 11 12name A2 A6 A3 C5 C3 C1plane (111) (111) (111) (111) (111) (111)direction [011] [110] [101] [110] [101] [011]
Schmid factor 15
√32
15
√32
√6
5 − 15√
6−1
5
√23 −1
5
√32
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 17 / 44
FE calculation using single crystal models Playing with Schmid Factor
Simple tension for a single crystal specimen
tensile test in the direction [123] 1 active slip systemSchmid factor Ms = (ns .t) (ms .t)
number 1 2 3 4 5 6name B4 B2 B5 D4 D1 D6plane (111) (111) (111) (111) (111) (111)direction [101] [011] [110] [101] [011] [110]
Schmid factor√
67
√6
14
√6
1417
√23
57√
6
√6
14
number 7 8 9 10 11 12name A2 A6 A3 C5 C3 C1plane (111) (111) (111) (111) (111) (111)direction [011] [110] [101] [110] [101] [011]
Schmid factor 17
√23
√6
747
√23 0 0 0
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 18 / 44
FE calculation using single crystal models Single crystal notched sample
Notched sample
A notched plate made of a single crystal is computed withZ-SeT/Zebulon. This example is to show the various possibilities of somemodels built in the code. You can play with two meshes:
notch1.geof has one layer and 60 elements
notch2.geof has two layers and 120 elements
You ca, go into these files and change c3d20 in c3d20r (reducedintegration for saving CPU time and storage space)The purpose of the exercice is to check various crystallographic potentialto make a global model with various slip system families. So, you can editcristal.mat, which is the material file, and choose the type of slip systemsyou wish. You can take several families. The only thing you have to checkis to use different names for the various families you introduce (the nameaf the end of the **potential line).
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 20 / 44
FE calculation using single crystal models Single crystal notched sample
Notched sample
A local framework is introduced in the problem notch2, so that the crystalaxes are rotated with respect to the specimen axes. You can define therotation you wish after the *rotation instance, in the ***materialsection.Now you can either:
Edit notch1.inp
Run notch1
Check notch1 results
or
Edit notch2.inp
Run notch2
Check notch2 results
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 21 / 44
Contents
1 Introduction
2 A crystal plasticity model
3 FE calculation using single crystal modelsPlaying with Schmid FactorSingle crystal notched sample
4 FE calculation using polycrystal models
5 Crystal plasticity computations on real microstructuresPolycrystalline β Ti under tensionSimulation of coherent diffraction in a polycrystalline film
FE calculation using polycrystal models
Objective
The purpose of this section is to:
Study the equations of some popular polycrystalline models
Check the source code for a few simple cases
Compute some typical loadings and evaluate yield surfaces
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 23 / 44
FE calculation using polycrystal models
Overview
Grain orientation is taken into account to define phases
Local stresses and strains are defined for each phase (concentrationrule)
The constitutive equations are written for each grain: slip systems(also twinning, etc...) are described on this level, by means of acollection of linear yield functions (Schmid law)
The global behavior is obtained as the average of each individualcontribution (homogenization)
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 24 / 44
FE calculation using polycrystal models
Ingredients of mean field theory
Macroscopic quantitiedΣ∼ ,E∼,E∼
p
Mean values per phaseσ∼
g , ε∼pg , γsg
Taylor modelε∼
pg = E∼p =⇒ σ∼
g 6= Σ∼
Static modelσ∼
g = Σ∼ =⇒ ε∼pg 6= E∼
p
Self–constitent estimates
σ∼g 6= Σ∼ , ε∼
pg 6= E∼p
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 25 / 44
FE calculation using polycrystal models
Self–consistent scheme
[Kroner, 1961]H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 26 / 44
FE calculation using polycrystal models
Scale transition rules
Secant and tangent approximations
Σ∼ = L∼∼eff : E∼
Lin and Kroner estimates (' Taylor)
σ∼ = Σ∼ + µ(E∼p − ε∼
p)
Berveiller and Zaoui’s estimate (1979)
σ∼ = Σ∼ + µα(E∼p − ε∼
p)
elastoplastic accommodation coefficient α varying from 1 to 0.01during deformation
limitations : elastoviscoplasticity, radial and monotonic loading, noreal grain morphology, limited interaction between grains, estimationthat may be out of the theoretical bounds, difficult to estimate strainheterogeneities inside the grains. . .
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 27 / 44
FE calculation using polycrystal models
Identification of scale transition rules
E∼ = E∼e + E∼
p, Σ∼ = C∼∼: E∼
e , Σ∼ =
Ng∑g=1
fg σ∼g , E∼
p =
Ng∑g=1
fg ε∼pg
σ∼g −→ ε∼
pg according to the constitutive equations of single crystalplasticityExample of efficient parametrized scale transition rule:
σ∼g = Σ∼ + C
(B∼ − β
∼g)
, B∼ =
Ng∑g=1
fg β∼
g , β∼
g= ε∼
pg − D εpgeq β
∼g
intergranular plastic accommodation variables β∼
with the nonlinearevolution law
scale transition parameters : C ,D 6= material parameters
identification procedure, inverse approachH. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 28 / 44
FE calculation using polycrystal models
Yield surfaces
The purpose is to build the initial and subsequent yield surfaces in anaxial-shear plane. A tensile loading is applied. You can work on thefollowing files:
the input file poly shear.inp edit – reset
the material file polyc.mat edit – reset
the grain file pp40 edit – reset
You can change the values of the coefficients in the material file, andcheck the change in shape:
RUN simulation
DRAW results
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 29 / 44
FE calculation using polycrystal models
Graphical illustration
This page is just to get a physical image of the active slip systems duringa tension. This is the opportunity to review the meaning of a RVE typecalculation (a special element with no nodes, and only one Gauss point).So, you can:
CHECK the input file (loading, BC, and material)
RUN the calculation
POST-PROCESS the slip systems
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 30 / 44
Contents
1 Introduction
2 A crystal plasticity model
3 FE calculation using single crystal modelsPlaying with Schmid FactorSingle crystal notched sample
4 FE calculation using polycrystal models
5 Crystal plasticity computations on real microstructuresPolycrystalline β Ti under tensionSimulation of coherent diffraction in a polycrystalline film
Crystal plasticity computations on real microstructures Polycrystalline β Ti under tension
Raw image reconstructed from DCT
raw image 302 × 302 px2 – slice 0/446
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 33 / 44
Crystal plasticity computations on real microstructures Polycrystalline β Ti under tension
Computation under tensionmesh from DCT image of ti sample with 130 grains [Ludwig et al., 2009]:
x y
z
x y
z
300
330
360
390
420
450
480
510
540
570
600
630
sigm
ises
map
:1.0
0000
ti
me:
1
min
:120
.328
max
:264
1.54
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 34 / 44
Crystal plasticity computations on real microstructures Simulation of coherent diffraction in a polycrystalline film
Numerical Diffraction Model
A 3 steps model (more in [Vaxelaire et al., 2010])
1 Generate 3D polycrystalline model
2D Voronoi cell generation in (x , y) planeextrusion along z to simulate columnar grains〈111〉 texture with random in plane orientation
2 Compute displacement field u(x , y , z)
Z-SeT/ZeBuLoN software suiteCubic ElasticityParallel computation for large meshes
3 Carry out Fourier Transform of exp (ıG.u(r))transfer u field on a regular gridcomplex FFT using fftwlibrary [Frigo and Johnson, 2005]now available as a post processingroutine within Z-SeT
x
y
z
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 40 / 44
Crystal plasticity computations on real microstructures Simulation of coherent diffraction in a polycrystalline film
Model parameters
physical parameters
50 grains
film dimensions: 500× 500× 50 µm3
cubic elasticity with C11 = 192 340 MPa, C12 = 163 140 MPa andC44 = 41 950 MPa
gold crystal atomic spacing a = 0.408 nm
mesh parametersm5 m3 m2 m1 m0
number of elements 1172 5940 27430 224410 1820200elements in grain 06 32 93 460 3860 31290elements in grain 39 12 45 160 1340 10880parallel computation no no no yes yes
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 41 / 44
Crystal plasticity computations on real microstructures Simulation of coherent diffraction in a polycrystalline film
Shape of the illuminated grain
x
y
z
grain 39 〈100〉
grain 06 〈111〉
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 42 / 44
Crystal plasticity computations on real microstructures Simulation of coherent diffraction in a polycrystalline film
Evolution of pattern with mesh density
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 20 40 60 80 100 120 140 160 180
disp
lace
men
t U1
Distance along grain boundary S
m0m1m2m3m5
maillage m5 maillage m3 maillage m2 maillage m1
gra
in06
oriente
〈111〉
qx
qy
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 43 / 44
Crystal plasticity computations on real microstructures Simulation of coherent diffraction in a polycrystalline film
Influence of the strain level
normalised 〈111〉 reflections (contours represent 0.3, 0.5, 0.7 and 0.9 values)
0.2% strain 0.6% strain 1.0% strain
0.2% strain 0.6% strain 1.0% strain
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 44 / 44
Crystal plasticity computations on real microstructures Simulation of coherent diffraction in a polycrystalline film
Frigo, M. and Johnson, S. G. (2005).
The design and implementation of FFTW3.
Proceedings of the IEEE, 93(2):216–231.
Special issue on “Program Generation, Optimization, and PlatformAdaptation”.
Ludwig, W., King, A., Reischig, P., Herbig, M., Lauridsen, E., Schmidt, S.,Proudhon, H., Forest, S., Cloetens, P., du Roscoat, S. R., Buffiere, J.,Marrow, T., and Poulsen, H. (2009).
New opportunities for 3d materials science of polycrystalline materials at themicrometre lengthscale by combined use of x-ray diffraction and x-rayimaging.
Materials Science and Engineering: A, 524(1-2):69–76.
Vaxelaire, N., Proudhon, H., Labat, S., Kirchlechner, C., Keckes, J., Jacques,V., Ravy, S., Forest, S., and Thomas, O. (2010).
Methodology for studying strain inhomogeneities in polycrystalline thin filmsduring in situ thermal loading using coherent x-ray diffraction.
New Journal of Physics.
H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 44 / 44