Multiscale Modeling of Indentation: from atom to continuum

Transcription

Multiscale Modeling of Indentation:
from atom to continuum
Hyung-Jun CHANG, Marc FIVEL, David RODNEY, Marc VERDIER,
SIMaP-GPM2, Grenoble INP / CNRS
[email protected]
Colloque Plasticité 2012, 11-13 Avril 2012, Metz
Multiscale Materials Modelling
Scanning Electron Microscope
Transmission Electron Microscope
Eye
Optical Microscope
High Resolution Electron Microscope
Discrete
dislocation
dynamics
Molecular
dynamics
Size [m]
Crystal plasticity
(polycrystals)
Crystal plasticity
(single crystals)
Time [s]
Homogenization technics
Continuum mechanics
Discrete Dislocation Dynamics (code TRIDIS)
Dislocations = edge and screw segments embedded in an elastic continuum
(Kubin, Canova and Bréchet 1992)
(similar to elastic inclusions)
Driving force = elastic stress tensors
(internal + applied)
v=
τb
B
r
b
τ
Example :
Frank Read source
directglide)
output
ExamplePlastic
: (112) deformation:
tensile test (double
Modern Discrete Dislocation Dynamics Codes
(e.g. Paradis)
(project NUMODIS (CNRS-CEA) L. Dupuy, M. Blétry)
Nodal code
Frank-Read source + partials
Dislocation junctions
Stacking Fault Tetraedra
Advantage : closer to reality
Drawback : Ten times slower
Eye
Optical Microscope
High Resolution Electron Microscope
Discrete
dislocation
dynamics
Molecular
dynamics
Size [m]
Crystal plasticity
(polycrystals)
Crystal plasticity
(single crystals)
Time [s]
Continuum mechanics
Eye
Example of scale transition :
Optical Microscope
From Electron
dislocation
dynamics to continuum mechanics
High Resolution
Microscope
Discrete
dislocation
dynamics
Molecular
dynamics
Size [m]
Crystal plasticity
(polycrystals)
Crystal plasticity
(single crystals)
Time [s]
Continuum mechanics
Identification of crystal plasticity constitutive equations
(Coll. L. Tabourot, C. Déprés, SYMME, Annecy)
γ&
(s)
asu ~ a = 0,09
1/m
⎛τ ⎞
= γ& ⎜ (s) ⎟
⎜τ ⎟
⎝ μ ⎠
(s)
(s)
0
τ = μb
(s)
μ
K = 32
12
(u)
a
ρ
∑ su
βR = yc ~ b
u =1
⎛
⎜
1⎜
ρ& (s) = ⎜
b⎜
⎜
⎝
12
∑ d su ρ (u)
u =1
K
⎞
⎟
⎟
− 2βRρ (s) ⎟ γ& (s)
⎟
⎟
⎠
3D tensorial framework
⎧
⎪
12
⎪
τ& μ(s) = ∑ ⎨
u =1 ⎪
2
⎪
⎩
μa su
12
∑a
p =1
sp
ρ
(p)
⎛
⎜
⎜
⎜
⎜
⎜
⎝
+ Recently revisited by : L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)
12
∑d
q =1
uq
K
ρ
(q)
⎞ ⎫
⎟ ⎪
⎟ (u) ⎪
(u)
− 2βRρ ⎟ γ& ⎬
⎟ ⎪
⎟ ⎪
⎠ ⎭
soit
12
τ& = ∑ h su γ& (u)
(s)
μ
u =1
Finite Element Implementation (ABAQUS : UMAT and VUMAT)
Example multicrystal Al
(Ph. D. Hyung-Jun CHANG (2009))
Example of DD applications:
(See also http://www.numodis.fr)
Clear channels in AISI 316L steel
Ph.D. Thomas NOGARET (2007)
Plastic behavior of BCC Fe
Ph.D. Julien CHAUSSIDON (2007)
Ph.D. Daniel GARCIA-RODRIGUEZ (2011)
Creep of ice single crystals
Ph.D. Juliette CHEVY (2008)
Micro-compression of Mg pillars
Ph.D. Gyu Seok KIM (2011)
Crack initiation in fatigue
Ph.D. Christophe DEPRES (2004)
Ph D. Chan Sun SHIN (2004)
(111) Nanoindentation in Cu single crystal
Ph.D. Hyung Jun CHANG (2009)
3D simulation of nanoindentation
Example #1 : From MD to DD
DD coupled to FEM = ideal tool to investigate nanoindentation issues :
indentation size effect, pile-up vs sink in mechanism, microstructure formation,…
Problems :
I- Coupling DD to FEM
- Superposition principle
[σ]eff = [σ]disl + [σ]FE
II- Need of a nucleation
criterion for dislocations:
1- What to put in ?
2- When to put it in ?
MD or experiments
Problem I: Enforcing boundary conditions :
DD-FEM coupling method (superposition)
E. van-der-Giessen, A. Needleman, Mater. Sci. Eng. , (1995)
Full problem
Fap
d2Ω
σ(M)
Ω
d1Ω
DD sub-problem
FE sub-problem
Fap-FD
FD
σ(M)
Ω
UD
Uap
Dislocation theory
(∞ medium)
~
σ = σ$ + σ
σ(M)
Ω
Uap- UD
Coupling with Finite Elements
(CAST3M)
Problem I: Enforcing boundary conditions :
DD-FEM coupling method (superposition)
Application to nanoindentation
Uap- UD
0-FD
Problem II: Nucleation criterion : What to introduce ?
(Molecular Dynamics modeling)
Simulation campaign :
Material = Ni (EAM potential)
Indenter = spherical repulsive potential
Monitored in displacement
2 radii : Rind = 60Å and 120 Å
3 sizes : 174x198x163 Å3 (521.640 atoms)
224x284x285 Å3 (1.675.080 atoms)
301x301x200 Å3 (1.524.600 atoms)
(111)
(121)
Periodic BC
Molecular Dynamic simulations of (111) indentation
The loading curves : study of the elastic part
4 E 2 ⎛ h
F=
Rind ⎜⎜
Hertz prediction :
3 1 −ν
⎝ Rind
Atomistic results :
F = αh p
Where
And
2.5e-006
1,65 < p < 1,75
20,15< α < 30,65
Ni1
Ni2
Ni3
Ni4
Ni6
Ni8intel
Ni8gnu
2e-006
1.5e-006
⎞
⎟⎟
⎠
3/ 2
(p decreases as Zmax increases)
(when F in nN and h in Å)
(111)
First Pop in
(121)
F[N]
Zmax
1e-006
5e-007
0
0
5e-010
1e-009
h[m]
1.5e-009
2e-009
The dislocation structure : study of the initial defect
The dislocation structure : structure at equilibrium
2 to 3 prismatic loops
High sensibility to boundary conditions
Molecular Dynamic simulations
The dislocation structure : structure after the first pop in
Same process = more prismatic loops with larger size
Horizontal half loops propagate to accommodate the indentation print
Dislocation nucleation criterion : What to put in ?
Atomistic simulations summarized :
;
Contact area
(111)
[110]
b3=[110]
Prismatic loops
Problem II: Nucleation criterion :
What to put in ?
When to put it in ?
Shape of Nucleation (MD,111)
Master curve (MD,111)
2.5
F [mN]
2.0
Ni1
Ni2
Ni3
Ni4
Ni6
Ni8
First Generation of
Dislocation loops
1.5
1.0
0.5
0.0
0.0
What : 3 Prismatic loops
0.5
1.0
depth [nm]
1.5
When : Follow master curve
Good : Criterion without any experimental results
Weak : MD response extrapolated for deeper indentation depth !
Alternative nucleation criteria :
- Experimental master curve
- Geometrically necessary dislocations
2.0
Dislocation Dynamics simulations
Specimen
Copper single crystal (111 surface)
Sphere (r=150nm)
Cone (angle = 71.2o)
Tip geometries
Nucleation method
1. Global criterion : Force controlled Nucleation
Master curve from MD (sphere) or Exp (cone)
2. GND criterion : Depth controlled Nucleation
Cross-slip probability
(effect of Temperature)
L δt ⎛ τ * −τIII ⎞
P =β
exp⎜
⎟
L0 δt0 ⎝ kT/ V ⎠
τ III = ∞(no ), 640 MPa(Hard ), 32MPa(Easy )
Dislocation Dynamics simulations
(111) Spherical indentation
Dislocation evolution (MD global crit. + no cross-slip)
5nm depth
10nm depth
(1 0- 1)
σ > line tension
Î F-R source
60nm depth
(1-2 1)
Contact area ↑
Î loop length ↑
(1 0- 1)
(1 0- 1)
(1-2 1)
(1-2 1)
Loop Length ↑
Line tension ↓
No space
to nucleation
(1 1 1)
(1 1 1)
(1-2 1)
Nucleation only
(similar to MD)
150nm
(1 1 1)
(1-2 1)
150nm
Nucleation and
Frank-Read sources
(1-2 1)
230nm
Frank-Read sources only
(111) Spherical indentation
Cross-slip effect (MD global crit.)
Before unloading (60nm depth)
230nm
230nm
230nm
No cross-slip
Hard cross-slip
Easy cross-slip
After unloading (60nm depth)
(1 0- 1)
(1 0- 1)
(1 0- 1)
230nm
(1-2 1)
(1-2 1)
(1-2 1)
230nm
230nm
Cross-slip↑ Î more irreversible micro structure
(111) Conical indentation : Force-Displacement response
(exp. Nucleation crit.)
Hardness
Total Force
0.5
MD global criterion
GND criterion
Exp. global criterion
30000
Hardness (MPa)
Force (mN)
0.4
35000
0.3
0.2
25000
MD global criterion
Exp. global criterion
GND criterion
20000
15000
10000
0.1
5000
0.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Total displacement(µm)
( μm)
MD global crit.
GND crit.
Exp. global crit.
0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
Indenting depth(µm)
(nm)
ÎTwo phase behavior
( linear to parabolic)
Hardness : Decreases with depth
Î single behavior
( ≈ linear response)
Exp. global crit. Î long range decreasing
Indentation Size Effect
3D simulation of nanoindentation
Time [s]
Exemple #2 : From Dislocation Dynamics to Continuum Mechanics
Size [m]
Experimental data
Sample preparation
Crystal ILL : (110), (001), (111) surface orientation
Cut by spark erosion from bulk single crystal (high ρini)
(110)
(001)
(111)
Crystal B. : (123), (111low) surface orientation
Grown from high purity Cu using Bridgman technique (low ρini)
(12-3)
(-541)
(111 low)
- 4 surface orientations
- 2 initial dislocation densities for (111) orientation
Experimental data (conical indentation)
Indentation loading curves
1200000
60000
111
110
111
001
123
(111low)
40000
30000
20000
10000
0
0
110
111
111
001
123
1000000
Stiffness (N/m)
50000
Force ( μN)
S
500
1000
800000
600000
400000
200000
0
1500
Depth (nm)
0
500
1000
1500
Depth (nm)
90
900
Hardness ( MPa)
Area
Area( (μμm
m22) )
80
800
70
700
60
50
600
110
111
001
123
Orientation
Orientation
111low
500
110
111
001
123
Orientation
111low
Strong effect of initial dislocation density & Weak effect of orientation
Experimental data (conical indentation)
Surface morphologies (AFM)
(001) Surface
(110) Surface
μm
5μm
(123) Surface
μm
μm
5μm
5μm
(111) Surface (highρini)
(111) Surface (lowρini)
μm
μm
Surface morphology
strongly affected by
Surface orientation
5μm
5μm
To check by
FEM modelling
Crystal Plasticity Modeling
Nanoindentation procedure
FEM
Experiment
Sapphire conical tip
Rigid body tip (identical)
Tip angle = 71.2o
Tip
Indentation depth
1400 nm
hsphere=200 nm
4 Specimens
4 Cu single crystals
Crystal plasticity theory
5mm × 5mm
Strain rate control
Loading
condition
−2
&
ε = 5 ×10 / sec
5mm × 5mm
Velocity control
Dislocation density based model
1/m
γ& (s)
⎛ (s) ⎞
(s) ⎜ τ
= γ& 0 (s) ⎟
⎜τ ⎟
⎝ μ ⎠
τ = μb
(s)
μ
12
∑a
u =1
su
ρ (u)
⎛ 12
⎞
⎜ ∑ d su ρ (u)
⎟
⎟
1⎜
− 2βRρ (s) ⎟ γ& (s)
ρ& (s) = ⎜ u =1
b⎜
K
⎟
⎜
⎟
⎝
⎠
Crystal plasticity model
⎧
⎪
12
⎪
τ& μ(s) = ∑ ⎨
u =1 ⎪
2
⎪
⎩
μa su
12
∑a
p =1
sp
ρ
(p)
⎛
⎜
⎜
⎜
⎜
⎜
⎝
3D ABAQUS simulations
12
∑d
q =1
uq
K
ρ
(q)
⎞ ⎫
⎟ ⎪
⎟ (u) ⎪
(u)
− 2βRρ ⎟ γ& ⎬
⎟ ⎪
⎟ ⎪
⎠ ⎭
soit
12
τ& = ∑ h su γ& (u)
(s)
μ
u =1
Parameters used in the crystal plasticity model
Elastic properties for T* = CE [E* ]
From text book
Initial dislocation density
and
Surface orientation
C11
168.4 GPa
C12
121.4 GPa
C44
75.4 GPa
Surface
Orientation
FWHM (111)
(θ 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
14/m2
1.20×10
Unknown
(123)
0.2
1
3.00×1013/m2
(111low)
0.08
0.2 ~ 0.3
6.00×1012/m2
From X-ray results
Hardening parameters
b
From DD theory
α1~6
a1~6
K
2.56×10-10 m
yc
1.43×10-9 m
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)
Normal
(0.01, 0.4, 0.4, 0.75, 1.0, 0.4), K=36
Same
a1-6 same as α1~6 , K=36
High K
(0.01, 0.4, 0.4, 0.75, 1.0, 0.4), K=100
M. Fivel, PhD Thesis, (1997) L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)
Effect of orientation
(001) Surface
(Hetero / Same)
(110) Surface
3%
5μm
5μm
(ρini= 1.2×1013/m2)
(111) Surface
(123) Surface
Weak effect on loading curve
5μm
5μm
Strong effect on surface displacement
Crystal Plasticity Modeling / Comparison with experiments
Effect of dislocation density for (111) orientation
(Hetero / Same)
60000
Force ( μN)
50000
40000
(111) Surface (high ρini)
111 surface orientation
Experiment
high density
low density
Simulation
14
2
ρini=1.8x10 /m
14
2
13
2
13
2
12
2
12
2
ρini=1.2x10 /m
30000
ρini=3.6x10 /m
ρini=1.2x10 /m
20000
ρini=6.0x10 /m
ρini=1.2x10 /m
Crystal ILL
Crystal B.
5μm
10000
0
0.0
0.2
0.4
μm
0.6
0.8
1.0
1.2
1.4
depth ( μm)
(111) Surface (low ρini)
μm
Strong effect on loading curve
Optimized initial dislocation density
ρini= 6.0×1012/m2 for (111low)
ρini= 1.2×1014/m2 for (111high)
What about the other orientations ?
5μm
Strong effect on surface shape
Comparison of quantitative results
EXP.
Force ( μN)
50000
40000
30000
ρini= 1.56×1014/m2
ρini= 1.20×1014/m2
ρini= 1.20×1014/m2
ρini= 3.00×1013/m2
ρini= 6.00×1012/m2
FEM
(110)
(111)
(001)
(123)
(111low)
(110)
(111)
(001)
(123)
(111low)
20000
1000000
Stiffness (N/m)
60000
800000
600000
400000
0
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0
200
400
Hardness ( MPa)
70
60
110
111
800
1000 1200 1400
Experiment
Simulation
900
Experiment
Simulation
80
600
Depth (nm)
depth ( μ m )
90
Area ( μm2)
exp sim
110
111
001
123
200000
10000
50
(hetero / same)
001
123
Optimized
initial densities
700
600
500
111low
Orientation
800
x5
2
(011)
1.56×1014/m
(111)
2
1.20×1014/mx4
(001)
2
1.20×1014/mx4
110
111
001
123
111low
Orientation
(123)
(111low)
3.00×1013/m21
2
6.00×1012/mx.2
; X-Rays
Effect of hardening parameters
for (111) orientation
70000
EXP
60000
α1~6
Hetero
μm
Normal
5μm
5μm
μm
40000
a1~6
Force ( μN)
50000
μm
Experiment
FEM
Hetero Taylor
normal
same
high K
Taylor
30000
20000
Same
10000
5μm
0
0.0
5μm
0.2
0.4
0.6
0.8
1.0
1.2
5μm
1.4
depth ( μm)
μm
Weak effect
on loading
curve
Obtained
hardening
parameter
Hetero + Same
High K
Need to check for other orientation
5μm
5μm
Influence surface morphology
Comparison of Surface morphology (hetero / same)
FEM
FEM
EXP
(001) Surface
(111) Surface (high ρini)
μm
5μm
5μm
5μm
5μm
5μm
μm
5μm
(111) Surface (low ρini)
μm
(110) Surface
EXP
5μm
μm
5μm
μm
(123) Surface
Confirmed hardening parameter
5μm
5μm
b
2.56×10-10 m
α1~6
Hetero
(0.122, 0.122, 0.07, 0.137, 0.122,0.625)
a1~6
K
Same
same as α1~6 , K=36
yc
1.43×10-9 m
Comparison of pile up morphology
(hetero / same)
0.4
Unload FEM
Unload EXP
111 (highρini)
0.2
0.1
0.0
0.0
45
90
135
180
225
270
315
360
0
45
90
135
Theta
θ
180
225
270
315
360
Theta
0.4
0.4
0.4
Unload FEM
Unload EXP
001
Uz ( μm)
0.2
0.3
0.3
0.2
0.2
0.1
0.1
0.1
0.0
0.0
0.0
0
45
90
135
180
Theta
225
270
315
360
0
45
90
135
Unload FEM
Unload EXP
110
Uz ( μm)
Unload FEM
Unload EXP
0.3
Uz ( μm)
111 (lowρini)
0.2
0.1
0
Unload FEM
Unload EXP
0.3
Uz ( μm)
Uz ( μm)
0.3
0.4
180
Theta
225
270
315
360
0
45
90
135
123
180
225
270
315
Theta
The hardening parameter (hetero/Same) is confirmed quantitatively
360
Conclusion : best set of parameters
Elastic
[ ]
T =C E
*
E
*
Initial dislocation density
and
Surface orientation
C11
168.4 GPa
C12
121.4 GPa
C44
75.4 GPa
Surface
Orientation
Relative disl.
density
Initial density
(total)
(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
Hardening
b
2.56×10-10 m
α1~6
Hetero
(0.122, 0.122, 0.07, 0.137, 0.122,0.625)
a1~6
K
Same
same as a1~6 , K=36
yc
1.43×10-9 m
L. Kubin, B. Devincre and T. Hoc, Acta Mater., (2008)
5 nm
Molecular Dynamics
50nm
1
2
3
4
5
6
Multi scale modelling of indentation
7
8
9
10
Dislocation dynamics
Finite Element Crystal plasticity
11
12
Experiments

Multiscale Modeling of Indentation: from atom to continuum

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