Lecture Micromechanics texture SFB 761

Why does a crystal rotate ?Why does a crystal rotate ?Dierk Raabe

Düsseldorf, Germany

[email protected]

SFB Class 2012

OverviewOverview

Roters et al. Acta Materi.58 (2010) 1

Some dislocation kinematics and kinetics: phenomenaSome dislocation kinematics and kinetics: phenomena

true strain

stress

3

Plastic deformation of a single crystal by dislocation slipPlastic deformation of a single crystal by dislocation slip

4


5


6


bvtZ

bXxn

t md1d

dd

7


S h id f tSchmid factor(orientation factor for that slip system)

8

Boundary condition: determines lab frame constraintsBoundary condition: determines lab frame constraints

11

Single crystal plasticity: crystal shear and crystal rotationSingle crystal plasticity: crystal shear and crystal rotation

12


bvtZ

bXxn

t md1d

dd

13

u=u(x y z)

Kinematics, displacementKinematics, displacement

u u(x,y,z)

1 2

(x(1),y,z) (x(2),y,z)

1 2

u(1)(x,y,z) u(2)(x,y,z)u (x y z)=u (x y z)

1 2

u(1)(x,y,z)=u(2)(x,y,z)

1 2

14

u=u(x y z)

Kinematics, displacementKinematics, displacement

u u(x,y,z)

1 2

(x(1),y,z) (x(2),y,z)

1 2

u(1)(x,y,z) u(2)(x,y,z)u (x y z)≠u (x y z)

1 2

u(1)(x,y,z)≠u(2)(x,y,z)

1 2

15

1 2

Kinematics, displacement, displacement gradient: generalKinematics, displacement, displacement gradient: general

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Kinematics: Micro-to-macro-transitionKinematics: Micro-to-macro-transition

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Geometrical interpretationGeometrical interpretation

?

18

Geometrical interpretationGeometrical interpretation

19

Complex boundary conditionsComplex boundary conditions

one dislocationmesoscopic boundary conditions(grain / orientation neighborhood)one dislocation

parallel loops

(grain / orientation neighborhood)

parallel loops

reactions

20

orientation change

Simplify boundary conditionsSimplify boundary conditions

Boundary conditions:

1) Upper bound treatment: iso-stress

2) Lower bound treatment: iso-strain2) Lower bound treatment: iso strain

21

sym sym sym sym

Iso-stress: single slip systemIso-stress: single slip system

a a

krit

b b

krit

d d

krit

c c

kritc b d a

m m m msym sym sym sym

1

D

33

T D33

T

a D

b

cD

kritaktiv

c

11

d

22

Single crystal plasticity: multiple slip (or twinning) systemSingle crystal plasticity: multiple slip (or twinning) system

mit

/

/mit

23

Single crystal yield surfaceSingle crystal yield surface

critijljlkik bana

1 crystal, 1 slip system: 33

slip system 1

same strain

jj

active)(1crits

..

33

differentstresses

11

slip system 2

active)(2crits

activecrit

critss bana

1 crystal, 2 slip systems:

11

ijljlkik bana

Ds=1

Iso-stress: multiple slip (or twinning) systemIso-stress: multiple slip (or twinning) system

33

Ds=2D

krit (+)aktiv)s1 (

..Vers

krit,(+) TBH

S

11

krit,(+)aktiv)s2 (

.

krit,(-)s2

krit,(-)aktiv)s1 (

.

VersDs=2

bcc fcc Bcc: 24 systems

Single crystal plasticitySingle crystal plasticity

kr

itbcc, fcc,

Section in stress space

Bcc: 24 systems

krit

krit krit

BCC, 48 systems

kr

it

krit

26



1) Upper bound treatment: iso-stress

2) Lower bound treatment: iso-strain2) Lower bound treatment: iso strain

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The Taylor ModelThe Taylor Model

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Crystal yield surface, Taylor Bishop-HillCrystal yield surface, Taylor Bishop-Hill

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Crystal yield surface, Taylor Bishop-HillCrystal yield surface, Taylor Bishop-Hill

Many crystals many slip systems:

33grain 1

Many crystals, many slip systems:

grain 2

grain 3g

grain 4

imposed strain

11

3% 8%

Homogeneity and boundary conditions – meso-scaleHomogeneity and boundary conditions – meso-scale

3% 8%

15%



1) Upper bound treatment: iso-stress (strain not compatible)

2) Lower bound treatment: iso-strain (forces not in equilibrium)2) Lower bound treatment: iso-strain (forces not in equilibrium)

32

Multiscale crystal plasticity FEM or FFTMultiscale crystal plasticity FEM or FFT

33Raabe, Zhao, Park, Roters: Acta Mater. 50 (2002) 421

Crystal Mechanics FEM, grain scale mechanics (2D)Crystal Mechanics FEM, grain scale mechanics (2D)

Experiment (DIC, EBSD)v Mises strain

Simulation (C )(CP-FEM)

v Mises strain

SachtleberSachtleber, , Zhao, Raabe: MaterZhao, Raabe: Mater. . Sc. Sc. EnginEngin. A 336 (2002) 81. A 336 (2002) 81 34

Lecture Micromechanics texture SFB 761

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Transcript of Lecture Micromechanics texture SFB 761