td Éléments finis cristallins

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

mailto:[email protected]

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


Contents

1 Introduction





Introduction

Evidences of crystallographic slip

Deformed pure polycrystalline copper (Nomarski contrast)


Introduction

Z-SeT/Zebulon

get the software: http://www.nwnumerics.com/Downloads/


http://www.nwnumerics.com/Downloads/

Contents

1 Introduction





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)



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



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



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


Contents

1 Introduction





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]



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





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





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






Schmid factor√

65

15

√32

15

√32

15

√23

15

√32

15√

6


Schmid factor 15

√32

15

√32

√6

5 − 15√

6−1

5

√23 −1

5

√32




tensile test in the direction [123] 1 active slip systemSchmid factor Ms = (ns .t) (ms .t)


Schmid factor√

67

√6

14

√6

1417

√23

57√

6

√6

14


Schmid factor 17

√23

√6

747

√23 0 0 0


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).


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


Contents

1 Introduction





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



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)



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



Self–consistent scheme

[Kroner, 1961]H. Proudhon (MINES ParisTech) TD Elements Finis cristallins Ecole MECANO 15-19 Mars 26 / 44


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. . .



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


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



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


Contents

1 Introduction





Crystal plasticity computations on real microstructures Polycrystalline β Ti under tension

Raw image reconstructed from DCT

raw image 302 × 302 px2 – slice 0/446


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


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



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



Shape of the illuminated grain

x

y

z

grain 39 〈100〉

grain 06 〈111〉



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



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



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.


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