1 Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics Hyung–Jun CHANG 9 th October, 2009 Supervisors Marc Fivel,

1

Analysis of nanoindentation size effectbased on

Crystal Plasticity and Dislocation Dynamics

Hyung–Jun CHANG

9th October, 2009

Supervisors

Marc Fivel, Marc Verdier (Grenoble INP)

Laurent Tabourot (Univ. de Savoie)

Heung Nam Han, Kyu Hwan Oh (Seoul Natl’ Univ.)

Samuel Forest (Centre des matériaux)

2 / 49

Introduction Macro scale Micro scale Discussion & Prospective

Principle of indentation

hf

∞∞

Specimen

Indenter

Pile upSink in

∞∞

h

P

Loading state

Unloading state

Measurement

1. Continuous measure : Force, Depth, Stiffness

2. Post measurement : surface imprint (contact area)

Objective : Obtain material property using the measurements

Macro scale Micro scale

3 / 49


Work done by L. Charleux (ideal case) For - Isotropic material : amorphous or poly crystal with fine grain

- Self symmetric indenter : conical or pyramidal

Mechanical parametersIndentation testModel analysis (2D FEM)

Surface imprint

P

0 h

S

Indentation curve

L. Charleux, ph.D thesis, Grenoble INP (2006)

E

ε

σy

00 εy

σ

Mechanical parametersy and E Hardness

H

0 h

Obtain - two mechanical parameters - stable hardness

4 / 49


Indentation with single crystal metal

Surface imprint Mesured hardness

Y.Y. Lim and M. M. Chaudhri, Philosophical Magazine A (1999) Work done by M. Verdier (2007)

: Anisotropic slip line : Indentation size effect

Need 3D modeling for anisotropic behavior and indentation size effect

With Self symmetric indenter : conical or pyramidal

5 / 49


Simulation model for indentation

Isotropic model (2D, axisymmetric)

anisotropic model (3D,continuum)

: Only for ideal (isotropic) indentation

: Constitutive law for anisotropy crystal plasticity model

anisotropic model (3D,discrete)

: Dislocation dynamics coupled with FEM

6 / 49


Reproducing experiment(single crystal, CPFEM)Reproducing experiment(single crystal, CPFEM)

Theoretical analysis (size effect, DD&CPFEM)

Sphere

Cone

Pyramidal, cube

Single crystal

Scale of indentation

Simulation(analysis tool)

Indentation

Scale

Experiment Micro-Indentation Nano-Indentationmm m nm Å

Indenters Sphere

Cone

Specimen Poly crystal

Single crystal (Copper)

FEM (isotropy)

FEM (anisotropy)

FEM (size effect)

DD

MD

FEM (anisotropy)

Multiscale method ( DD base )

FEM (size effect)

Nano-Indentation

Theoretical analysis (ideal, FEM)

Theoretical analysis (initiation, MD)

Vickers Berkovich Cone

sphere

MD

Theoretical analysis (initiation+evolution, DD)

7 / 49


Contents

• Part 0 : Introduction

• Part I : macro scale : experiments and FEM– Experiments– Crystal plasticity modeling– Finite element simulations

• Part II : micro scale : DD simulation– Simulation method– DD simulation of spherical indentation– DD simulation of conical indentation

• Discussion & prospective– FEM modeling (size effect)– DD modeling (Size effect, local rotation)

8 / 49


Contents





Continuum modeling

9 / 49


Nanoindentation procedure

Tip

4 Specimens

Loading condition

Tip

radi

us =

3.3

m

hsphere=200 nm

4 Cu single crystals

Experiment

Tip angle = 71.2o

Sapphire sphero-conical tip

5mm × 5mm

Strain rate control

sec/105 2

FEM

1400 nm

Indentation depth

Rigid body tip (identical)

5m × 5m

Velocity control

Crystal plasticity theory

• Objective : To check– Orientation effect

– Dislocation density effect

• Mean – Force, Stiffness, Contact area and

Hardness – Surface displacement

(symmetry, pile-up height)

Nanoindentation procedure (EXP)

- Experiment

10 / 49


Sample preparation for experiment

(110), (001), (111) surface orientation

(123), (111low) surface orientation

Grown from high purity Cu using Bridgman technique

(001)

(12-3)

(-541)

(111 low)

Cut by spark erosion from bulk single crystal (high ini)

(low ini)

- 4 surface orientations

- Experiment

(111)(110)

- 2 initial dislocation densities on (111) orientation

11 / 49


Quantitative results (EXP)

Strong effect of initial dislocation density & Weak effect of orientation

S

- Experiment

12 / 49


Surface morphologies (AFM)

5m

m

5m

m

m5m

m

5m

m

5m

(001) Surface (110) Surface (123) Surface

(111) Surface (highini) (111) Surface (lowini)

Surface orientation

Surface morphology strongly affected by

To check by FEM modelling

- Experiment

13 / 49


Nanoindentation procedure

Tip

4 Specimens

Loading condition

Tip

radi

us =

3.3

m

hsphere=200 nm

4 Cu single crystals

Experiment

Tip angle = 71.2o

Sapphire conical tip

5mm × 5mm

Strain rate control

sec/105 2

FEM

1400 nm

Indentation depth


5mm × 5mm

Velocity control


FEM

1400 nm

Indentation depth


5mm × 5mm

Velocity control


• Ingredient : Physical constitutive equations

– Dislocation density based model of crystal plasticity

• Procedure :– Inverse method to find the

best set of parameters

Nanoindentation procedure (FEM)

- CP model

14 / 49



Plasticity = activation of many slip systems

1 21 2

x x

x1 x2

01(C1) 02(C3) 03(C5) 04(A2) 05(A3) 06(A6)

(1 1 -1) [0 1 1] (1 1 -1) [1 0 1] (1 1 -1) [1 -1 0] (1 -1 -1) [0 1 -1] (1 -1 -1) [1 0 1] (1 -1 -1) [1 1 0]

07(D1) 08(D4) 09(D6) 10(B2) 11(B4) 12(B5)

(1 -1 1) [0 1 1] (1 -1 1) [1 0 -1] (1 -1 1) [1 1 0] (1 1 1) [0 1 -1] (1 1 1) [1 0 -1] (1 1 1) [1 -1 0]

12 slip systems for Copper single crystal

- CP model

15 / 49


Constitutive law for crystal plasticity

L. Tabourot, M. Fivel, E. Rauch, Mater. Sci. Eng. A (1997)

*E* ECT Elastic

Equations Key variable

CE

ini , sp

asp, Kg

+Surface

orientation

sm

s

sss sign

r

1

0 s0

* μT :sFlow law

12

1p

ps b spα

ssc

g

p

p

s yKb

21

12

1

spa

Hardening law

gK

spa

spα

- CP model

Evolution law

16 / 49


Obtaining variables

Surface Orientation

FWHM (111)( scan °)

Relative disl. density

Initial density(total)

(011) 1.35 6 ~ 8 1.56×1014/m2

(111) 0.57 3 ~ 4 1.20×1014/m2

(001) 0.56 3 ~ 4 1.20×1014/m2

(123) 0.2 1 3.00×1013/m2

(111low) 0.08 0.2 ~ 0.3 6.00×1012/m2

Initial dislocation density and

Surface orientation

From X-ray results

C11 168.4 GPa

C12 121.4 GPa

C44 75.4 GPa

b 2.56×10-10 m yc 1.43×10-9 m

1~6

Taylor (0.09, 0.09, 0.09, 0.09, 0.09, 0.09)

Hetero (0.122, 0.122, 0.07, 0.137, 0.122,0.625)

a1~6

K

Normal (0.01, 0.4, 0.4, 0.75, 1.0, 0.4), K=36

Same a1-6 same as 1~6 , K=36

High K (0.01, 0.4, 0.4, 0.75, 1.0, 0.4), K=100

Elastic properties for

From literatures

Hardening parameters

From DD theory

L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)M. Fivel, PhD These, (1997)

*E* ECT

Unknown

- CP model

17 / 49


Implementation in ABAQUS (UMAT / VUMAT) & Validation

Well know three stage curve reproduced

(Taylor / normal)

- CP model

18 / 49


Application to nanoindentation Effect of orientation

5m 5m

5m 5m

(001) Surface (110) Surface

(123) Surface(111) Surface

3%

(Hetero / Same)

Strong effect on surface displacement

ini= 1.2×1013/m2)

- CP FEM

Weak effect on loading curve

19 / 49


m(111) Surface (high ini)

5m

m(111) Surface (low ini)

5m

Effect of dislocation density for (111) orientation

Optimized initial dislocation density

(Hetero / Same)

ini= 6.0×1012/m2 for (111low)

ini= 1.2×1014/m2 for (111high)

What about the other orientations ?

- CP FEM

Strong effect on surface shape

Strong effect on loading curve

20 / 49


Comparison of quantitative results (hetero / same)

Optimizedinitial densities

(011) 1.56×1014/m2 (123) 3.00×1013/m2

(111) 1.20×1014/m2 (111low) 6.00×1012/m2

(001) 1.20×1014/m2

- CP FEM

ini= 1.56×1014/m2 (110)

ini= 1.20×1014/m2 (111)

ini= 1.20×1014/m2 (001)

ini= 3.00×1013/m2 (123)

ini= 6.00×1012/m2 (111low)

21 / 49


m

5m 5m

m

5m 5m

m

5m 5m

Taylor Hetero

Normal

Same

High K

Effect of hardening parametersfor (111) orientation

Influence surface morphology

Obtained hardening parameter

Hetero + Same

5m

mEXP

Need to check for other orientation

- CP FEM

Weak effect on loading curve

22 / 49


Comparison of Surface morphologyFEM EXP FEM EXP

m

m

m

m

m

5m 5m

5m 5m

5m 5m

5m 5m

5m5m

(001) Surface

(110) Surface

(123) Surface

(111) Surface (high ini)

(111) Surface (low ini)

b 2.56×10-10 m yc 1.43×10-9 m

1~6 Hetero (0.122, 0.122, 0.07, 0.137, 0.122,0.625)

a1~6

KSame same as a1~6 , K=36

(hetero / same)

Confirmed hardening parameter

- CP FEM

23 / 49


Comparison of pile up morphology

111 (lowini)

001 110 123

111 (highini)

(hetero / same)

The hardening parameter (hetero/Same) is confirmed quantitatively

- CP FEM

24 / 49


Conclusion : best set of parameters

Initial dislocation density and

Surface orientation

C11 168.4 GPa

C12 121.4 GPa

C44 75.4 GPa

b 2.56×10-10 m yc 1.43×10-9 m

1~6 Hetero (0.122, 0.122, 0.07, 0.137, 0.122,0.625)

a1~6

KSame same as a1~6 , K=36

L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)

*E* ECT

Hardening

Elastic

- CP FEM

Surface Orientation

Relative disl. Density

(from X-ray)

Initial density(from FEM)

(011) 6 ~ 8 1.56×1014/m2

(111) 3 ~ 4 1.20×1014/m2

(001) 3 ~ 4 1.20×1014/m2

(123) 1 3.00×1013/m2

(111low) 0.2 ~ 0.3 6.00×1012/m2

25 / 49


Summary of EXP and FEM studies

• Effect of key variables– Orientation

• Weak effect to Force displacement curve• Strong effect on surface morphology (imprint symmetry)

– Initial dislocation density• Strong effect on Force displacement curve• Effect to pile up height

– Hardening parameters • Weak effect to Force displacement curve • Effect to details of surface morphology and pile up height

• Comparison between EXP and FEM with best parameter set Reasonable both qualitatively and quantitatively

• Limits– No size effect Dislocation dynamics

26 / 49


Contents





Discretemodeling

27 / 49


= +

FS

FTOT= FFEM+ FDD

FDD

FFEM SFEM SDD

DDDDE =DTOTDTOT

Bulk Bulk Infinite region

Virtual indenter

DTOT : Total indenting depthDDD : depth from dislocation fieldDE : depth from Elastic fieldSFEM : Elastic stress field

FTOT : Total applied loadFDD : Load from dislocation fieldFFEM : Load from Elastic fieldFS : Surface load from dislocation field

Coupling (Superposition) method

= +

FS-FS

FTOT= FFEM+ FDD

FDD

FFEM SFEM SDD

DDDDE =DTOT-DDD DTOT

Bulk Bulk Infinite region

Virtual indenter

DTOT : Total indenting depthDDD : depth from dislocation fieldDE : depth from Elastic fieldSFEM : Elastic stress field

FTOT : Total applied loadFDD : Load from dislocation fieldFFEM : Load from Elastic fieldFS : Surface load from dislocation field

100nm 100nm 100nm

Coupled FEM DD

E. van-der-Giessen, A. Needleman, Mater. Sci. Eng. , (1995)

- Simulation method

28 / 49


Algorithm of the coupled simulation

Dislocation dynamics (TRIDIS)

Finite Element Method (CASTEM)

Nucleate ?

Yes

No

Get dDD, dFEM, FS,DD from previous step

Impose B.C. at top Surface

Solve an elastic FEM problem

Update dDD, FS,DD

Nucleation (it could be failed)

Dislocation Dynamics Steps

Equilibrium ?

Over constraint ?

START CALCULATION

END CALCULATION

Yes

No

Yes

No

Nucleate ?

No

Yes Recalculate FEM ?

Yes

No

Update dDD, FS,DD

Remove over constraints

Need a criterion !

What (shape) ?When (condition) ?

Dislocation dynamics

- dislocation multiplication- dislocation nucleation ?

- Simulation method

MD simulations

29 / 49


Nucleation criterion #1 : MD global criterion

First Generation of Dislocation loops

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0

F [m

N]

depth [nm]

Ni1Ni2Ni3Ni4Ni6Ni8

Master curve (MD,111)Shape of Nucleation (MD,111)

When : Global criterion (Master curve)

Good : criterion without any experimental resultsWeak : MD cannot tell the force for deeper indentation depth

Need a nucleation criterion without master curve

What : 3 Prismatic loops

- Simulation method

30 / 49


Nucleation criterion #2 : GND criterion

Dislocation dynamics (TRIDIS)

Finite Element Method (CASTEM)

Nucleation(it could be failed)

Impose B.C. at top Surface

Solve an elastic FEM problem

Update dDD, FS,DD

Dislocation Dynamics Steps

Equilibrium ?

Over constraint ?

START CALCULATION

END CALCULATION

Yes

No

Yes

No

Remove over constraints

Get dDD, dFEM, FS,DD from previous step

Deformation and GND

Flow Chart

xindxDD,Nucle

nucleationDDind xx ,

W. D. Nix, H. Gao, J. Mech. Phys. Solids, (1998)

Nucleation imposed by indentation depth

Good : without any experimental results Good : without any master curves

- Simulation method

31 / 49


50nm 50

nm

Cone (angle = 71.2o)

Copper single crystal (111 surface)

Sphere (r=150nm)

Calculation Parameters

Tip geometries

Cross-slip probability PL

L

t

t kT VIII

0 0

exp/

*

EasyMPaHardMPanoIII 32,640,

Specimen

- Simulation method

Nucleation method Master curve from MD (sphere) or Exp (cone)

1. Global criterion : Force controlled Nucleation

2. GND criterion : Depth controlled Nucleation

: For indentation ?

32 / 49


Contents





Discretemodeling

- Spherical indentation (111)

33 / 49


Dislocation evolution (MD global crit. + no cross-slip)

(1 0- 1)

(1-2 1)

(1 1 1)

(1-2 1) 150nm

10nm depth5nm depth

(1 0- 1)

(1-2 1)

(1 1 1)

(1-2 1) 150nm

Nucleation only (similar to MD)

Nucleation andFrank-Read source

230nm

(1 0- 1)

(1-2 1)

(1 1 1)

(1-2 1)

60nm depth

Frank-Read source only

Loop Length ↑Line tension ↓

Contact area ↑ loop length ↑

> line tension

F-R source

No space to nucleation


34 / 49


Force displacement curveMD global crit. + no cross-slip GND crit. + no cross-slip

MD global crit. + Cross-slip condition

Cross-slip Deviation delayed

Nucleation allow

Nucleation forbidden


MD master curve

MD master curve

GND crit. closer to MD prediction

MD global crit.

GND crit.

MD master curve

35 / 49


100nm 100nm 100nm

No cross-slip Hard cross-slip Easy cross-slip

Cross-slip effect (30nm depth)

150nm

(1 1 1)

(1-2 1)

(1 0 -1)

(1-2 1)

150nm

(1 1 1)

(1-2 1)

(1 0 -1)

(1-2 1)

150nm

(1 1 1)

(1-2 1)

(1 0 -1)

(1-2 1)

Cross-slip↑ def. homogeneous ↑, Glissile loops ↓, Plastic zone size ↑


36 / 49


Cross-slip effect after unloading

230nm

(1 0- 1)

(1-2 1)

230nm

(1 0- 1)

(1-2 1)

230nm(1 0- 1)

(1-2 1)

Before unloading (60nm depth)

After unloading (60nm depth)

No cross-slip Hard cross-slip Easy cross-slip

230nm230nm

230nm

Cross-slip↑ more irreversible micro structure


37 / 49


Contents





Discretemodeling

- Conical indentation (111)

38 / 49


Nucleation criterion for conical indentation

Total Force Hardness

MD global crit. GND crit.

Exp. global crit. single behavior ( linear response)

Hardness : Drop down Stable

MD master c

urve, sphere

Exp. master curve, Berkovich


Two phase behavior ( linear to parabolic)

Exp. global crit. long range decreasing

39 / 49


Dislocation Structure (60nm depth)

GND crit. MD global crit. Exp. global crit.

230nm

230nm

(1 1 1)

(1-2 1)

(1 0- 1)

(1-2 1)

230nm

230nm

(1 1 1)

(1-2 1)

(1 0- 1)

(1-2 1)

230nm

230nm

(1 1 1)

(1-2 1)

(1 0- 1)

(1-2 1)

Prismatic loops & Glissile loops Prismatic loops & Junctions


40 / 49


Coplanar Junction : unexpected behavior !!

321 bbb

230nm

b3=(1 -1 0)

(1 1 2)

(1 -1 0)

b2=(1 0 1)b1=(0 1 1)

b1- b2

b3

3111 b

21,//111 bb

- Activate dislocations

perpendicular to loading direction

- Mobile dislocations

Immobile dislocations


Coplanar Junctions

41 / 49


Summary of DD results

• Dislocation evolution source : Size effect– Nucleation in size effect phase– Frank-Read source in no size effect phase– Cross slip– Coplanar junction (>50nm, Conical indenter with Exp. global crit.)

• Dislocation structures– Prismatic loops

• Increase by nucleation

– Glissile loops • Increase by Frank Read source• Decrease by Cross slip, Coplanar junction, Unloading

42 / 49


Contents





43 / 49


Problem of CPFEM model :Size effect

Force curve of (111) Surface

Curvature of EXP < Curvature of FEM

FEM(ini)

EXP

FEM(lower ini)

- FEM modeling (Size effect)

Problem !!! : Different ini for deeper indentation depth

Solution : Fit the curvature

How ? : Using Generalized plasticity theory (Ex. Strain gradient theory)

Access to size effect

44 / 49


Simplified strain gradient theory

Extra stress extra ↓ if indentation depth ↑

localpglobalavgp

SSD

SSDtotalL

Kb ,_,

12


Usual internal stress

- No extra parameter and extra degree of freedom- Easy to implement in FEM

= =

strain gradient theorydislocation theory

PROBLEM !! Very high calculation cost

Simplified theory (proposed)

: GND: SSD

45 / 49


First FEM results of simplified model

- reproduce indentation size effect, qualitatively- Need detailed verification


(001) Surface orientation

46 / 49


Origin of Size effect (Microscale)

• Dislocation structures beneath indenter– Size effect phase : Prismatic loops dominant– Stable hardening phase : Glissile loops dominant

Prismatic loops dominant Glissile loops dominant

(1 1 1)

(1-2 1) 150nm 230nm

(1 1 1)

(1-2 1)

Dislocation dynamics can help understanding size effect

- DD modeling (Size effect)

47 / 49


Link between size effect and local rotation

• Local rotation with scale (Simulation)– Microscale DD :

– Size effect theory :

– Crystal plasticity :

• Local rotation with scale (Experimental, EBSD)

κ : lattice torsion curvature

x

ffext

oldEnew GFG1

curlItrtl Tdd bα

M. Rester, C. Motz, R. Pippan, Acta Mater., (2008)

- DD modeling (local rotation)

ANR CATSIZE project with Samuel Forest (CMR) : Connenction bet. Cosserat & DD

48 / 49


Thank you for your attention

49 / 49


Contents





Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics

1 Analysis of nanoindentation size effect based on Crystal Plasticity and Dislocation Dynamics Hyung–Jun CHANG 9 th October, 2009 Supervisors Marc Fivel,

Documents

d thesis

indentationisotropic

axisymmetricanisotropic

ideal isotropic indentation

anisotropic behavior

anisotropic slip line

material property

samuel forest centre