Dierk Raabe Ab Initio Simulations In Metallurgy
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Transcript of Dierk Raabe Ab Initio Simulations In Metallurgy
MS&T Oct. 2008 Pittsburgh
Coupling Density Functional Theory with Continuum Mechanics for
Alloy Design
Coupling Density Functional Theory with Continuum Mechanics for
Alloy Design
D. Ma*, M. Friák, W. Counts, D. Raabe, J. Neugebauer
Max Planck Institute for Iron Research, Düsseldorf, Germany
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
Multi-scale Modeling
Å
nm
μm
mm
EH
bv
C
r
VF
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
Multi-scale Modeling
Å
nm
μm
mm
EH
bv
C
r
VF
Å mm
EH C
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
Two Examples:(1) β-Ti Alloys for
implants(2) Mg-Li Alloys for
lightweight structures
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
β-Ti alloy design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants
as input for CPFEM5. Summary
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as
Input for CPFEM5. Summary
MS&T Oct. 2008 Pittsburgh
1. Motivation
MS&T Oct. 2008 Pittsburgh
1. Motivation
M. Niinomi, Sci. Tech. Adv. Mater. 2003 M. Niinomi, Mater. Sci. Eng. 1998
Main challenges in designing the bone replacement:(1) Bio-compatibility(2) Reduce the elastic stiffness(3) Stabilize the β-phase
Ti-Nb binary system
~20GPa
~70GPa
>100GPa
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as
Input for CPFEM5. Summary
MS&T Oct. 2008 Pittsburgh
2. Phase Analysis
STHF )()1()()()( 11 XExTixEXTiEXTiH xxxx
vc SSS
))1ln()1()ln((B xxxxkS
DFT
MS&T Oct. 2008 Pittsburgh
wanted bcc-based phase that is softer but metastable
Nb
Tiunwanted hcp-based phase
that is stiffer and stableBCC structure of Ti-Nb alloy HCP structure of Ti-Nb alloy
2. Phase Analysis
MS&T Oct. 2008 Pittsburgh
2. Phase Analysis
MS&T Oct. 2008 Pittsburgh
2. Phase Analysis
MS&T Oct. 2008 Pittsburgh
XRDDFT
2. Phase Analysis
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as
Input for CPFEM5. Summary
MS&T Oct. 2008 Pittsburgh
Lattice constantsMinimum energyBulk modulus
3. Elastic Properties
Ab-initio calculation:Equilibrium lattice constants
MS&T Oct. 2008 Pittsburgh
00
02
10
002
1
)(4
3)( 1211
2 CCU tetr
)(2
312112
2
CCU tetr
3
2 1211 CCB
000
00
00
4422)( CU tri
442
2
4CU tri
Ab-initio calculation: Equilibrium elastic constants
ε, strain tensorδ, strainU, elastic energy densityB, bulk modulus
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
Ab initio calculation results of the elastic constants:
Composition C11 [GPa] C12 [GPa] C44 [GPa] Az EH [GPa]
Ti-18.75at.%Nb
131.2 114.5 26.8 3.210 49.4
Ti-25at.%Nb 143.6 125.9 21.4 2.418 44.2
Ti-31.25at.%Nb
154.8 118.5 19.2 1.058 54.9
C11, C12, C44: elastic stiffness constantsAZ, Zener‘s ratioEH: homogenized Young‘s modulus by Hershey‘s model
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
Ti-18.75at.%Nb Ti-25at.%Nb Ti-31.25at.%Nb
Az=3.210 Az=2.418 Az=1.058
[001]
[100] [010]
Young‘s modulus surface plots
The elastic properties of the Ti-Nb binary alloys become isotropic as the Nb content increases
Pure Nb
Az=0.5027
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
single-crystallineC11, C12, C44, B0
polycrystallineYoung modulus
“scale-jumping”
(across the meso-scale)macro-scale
micro-scale 64μ H4 + 16(4C11 + 5C12)μ H
3 + [3(C11+ 2C12) × (5C11+ 4C12) -8(7C11 – 4C12)C44]μ H
2 -(29C11 – 20C12)(C11+2C12)C44 μ H –3(C11 + 2C12)2(C11 – C12)C44 = 0
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
MECHANICALINSTABILITY!!
theory: bcc polycrystals
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
MECHANICALINSTABILITY!!
theory: bcc polycrystals
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
MECHANICALINSTABILITY!!
theory: bcc polycrystals
Ti-hcp: 117 GPa
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
MECHANICALINSTABILITY!!
Ultra-sonic measurement
exp. polycrystals !
bcc+hcp phases theory: bcc
polycrystals
Ti-hcp: 117 GPa
3. Elastic Properties
MS&T Oct. 2008 Pittsburgh
β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic constants as
input for CPFEM5. Summary
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM
Required input data of the materials properties in crystal plasticity finite element method
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4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM
Plane strain compression: (1) Influence of the elastic anistropy (2) predict the texture evolution
Bending test: Homogenized elastic
properties of textured and non-texture materials
MS&T Oct. 2008 Pittsburgh
4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM
Elastic constants of a single crystal
flow curve from the compression test on solution annealed Ti30at.%Nb
Random texture
The plastic property is kept, and only the elastic property is varied!!!
MS&T Oct. 2008 Pittsburgh
4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM
εh=0
εh=30%
εh=60%
εh=90%
φ1 (0°~90°)
Φ(0°~90°)
φ2=45°
γ-fiber
α-fiber
0° 90°
90°
0°
MS&T Oct. 2008 Pittsburgh
4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM
Elastic constants of a single crystal
Textured and non texture
MS&T Oct. 2008 Pittsburgh
4. Elastic Constants as Input of CPFEM4. Elastic Constants as Input of CPFEM
0
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Displacement [a.u.]
Loa
d [a
.u.]
Ti18.75Nb Non-TexturedTi18.75Nb TexturedTi25Nb Non-TexturedTi25Nb TexturedTi31.25Nb Non-TexturedTi31.25Nb Textured
Ti18.75Nb Non-TexturedTi18.75Nb TexturedTi25Nb Non-TexturedTi25Nb TexturedTi31.25Nb Non-TexturedTi31.25Nb Textured
MS&T Oct. 2008 Pittsburgh
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
β-Ti alloys design1. Motivation2. Phase analysis3. Elastic properties4. Elastic Constants
as Input of CPFEM5. Summary
MS&T Oct. 2008 Pittsburgh
1. Thermodynamic stability of hcp- and bcc-Ti was studied2. Configurational entropy at finite temperature stabilizes bcc Ti-
Nb phase3. Volume fractions have been calculated using the Gibbs
construction 4. Polycrystalline two-phase Young’s modulus has been
theoretically predicted employing the Hershey and CPFEM homogenization methods
5. Very good agreement between theoretical prediction and experiment
6. The calculated elastic constants (DFT) can be used as input for CPFEM
5. Summary
Nb SHOULD BE THE PRIMARY ALLOYING ELEMENTS IN Ti FOR HUMAN IMPLANT MATERIALS
MS&T Oct. 2008 Pittsburgh
Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the Elastic Properties4. Summary
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the Elastic Properties4. Summary
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
• Magnesium (and its alloys) are light weight and relatively strong
• Ideal lightweight structural material
• Magnesium (and its alloys) are generally hcp
• Not ductile, textures• Problematic for industrial
applications (anisotropy)
Magnesium Good Magnesium Bad
How can hcp magnesium be transformed into bcc/fcc magnesium?
1. Motivation1. Motivation
MS&T Oct. 2008 Pittsburgh
1. Motivation
Ultra light-weight structural material Li = 0.58 g/cm3 Mg = 1.74 g/cm3
hcpbcc
hcp+
bcc
MS&T Oct. 2008 Pittsburgh
1. Motivation
Ordered alloys (periodic structures) Ground state calculations (0 K)
11 different bcc alloys
Physical Limitations
Use DFT to find the bcc MgLi alloy composition with optimal elastic properties
Goal:
Calculate single crystal Cij’s
Homogenize to get isotropic polycrystal elastic constants
Analyze engineering ratio’s
MS&T Oct. 2008 Pittsburgh
Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the elastic Properties4. Summary
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
2. Elastic Properties: Bulk Modulus
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100
at. % magnesium
Bu
lk M
od
ulu
s (G
Pa)
Simulation: 0 K + Experiment: 298 K
Li
MS&T Oct. 2008 Pittsburgh
-10
-5
0
5
10
15
20
25
30
0 20 40 60 80 100
at. % magnesium
Sh
ear
Mo
du
lus
(G
Pa)
Voight & Reuss Bounds
Self Consistent Homgenization
Experiment
Experiment is reasonably well reproduced
Li dominate alloys are very soft
Optimal G (17 GPa)around bcc phase
boundary (70 at % Mg)bcc
Mg is unstable
HerG
ReussG
VoigtG
2. Elastic Properties: Shear Modulus
Li
MS&T Oct. 2008 Pittsburgh
2. Elastic Properties:Young‘s Modulus
-20
-10
0
10
20
30
40
50
60
0 20 40 60 80 100
at. % magnesium
Yo
un
gs
Mo
du
lus
(GP
a)
Voight & Reuss Bounds
Self Consistent Homgenization
Experiment
Li dominate alloys are very soft
Optimal E (45 GPa)around bcc phase
boundary (70 at % Mg) bcc Mg is
unstable
VoigtE
ReussE
HerE
Experiment is reasonably well reproduced
Li
MS&T Oct. 2008 Pittsburgh
2. Elastic Properties: Poisson‘s Ratio
0
0,1
0,2
0,3
0,4
0,5
0,6
0 20 40 60 80 100
at. % magnesium
Po
isso
n's
Rat
io
Voight and Ress Bounds
Self Consistent Homogenization
Experiment
Softer alloys have a higher n
Softer alloys have a lower n
Voigtν
Reussν
Herν
Experiment is reasonably well reproduced
Li
MS&T Oct. 2008 Pittsburgh
Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the elastic Properties4. Summary
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
3. Analysis of the Elastic Properties
Defined as
Based on experimental observationsMeasure of ductile vs. brittle behavior
G
B
ModulusShear
ModulusBulk
1.75 critical value B/G > 1.75 DUCTILE B/G < 1.75 BRITTLE
G Resisting Plastic Flow B Bond Strength
OppositionTo
Fracture
1.75 is more a transition zone
MS&T Oct. 2008 Pittsburgh
3. Analysis of the Elastic Properties
Stiffer bcc Mg-Li alloys Ductile/brittle transition region
1
1,5
2
2,5
3
3,5
4
4,5
0 20 40 60 80 100at. % Solute
B/G
AlLi; Solute:Li; DFT
LiMg; Solute:Mg; DFT
LiMg; DFT Error Est.
LiMg; Solute:Mg; Experiment
B/G = 1.75
ReussG
B
VoigtG
B 0.9
HerG
B
Brittle Region
Ductile Region
MS&T Oct. 2008 Pittsburgh
3. Analysis of the Elastic Properties
Defined as
Design Criteria Maximum stiffness for minimum weight
Typical Values (MPa m3/kg) Graphite Fiber 127.78 Graphite Fiber/epoxy 43.53 Steel 26.41 Aluminum 25.93 PET (polymer) 2.15 Lead 1.41
ρ
E
Density
Modulus sYoung'
MS&T Oct. 2008 Pittsburgh
3. Analysis of the Elastic Properties
Better than Al-Mg. Comparable to Al-Li.
ρ
EReuss
ρ
EVoigt
ρ
EHer
0
5
10
15
20
25
30
35
40
45
50
0 20 40 60 80 100at. % Solute
Sp
ecif
ic M
od
ulu
s (M
Pa
m3 /k
g)
AlLi; Solute Li; ExpAlMg; Solute Mg; ExpLiMg; Solute Mg; ExpLiMg; Solute Mg; DFTLiMg; DFT Bounds
MS&T Oct. 2008 Pittsburgh
3. Analysis of the Elastic Properties
MS&T Oct. 2008 Pittsburgh
Mg-Li alloy design1. Motivation2. Elastic properties3. Analysis of the elastic Properties4. Summary
Max-Planck-Institut für Eisenforschung, DüsseldorfMax-Planck-Institut für Eisenforschung, Düsseldorf
MS&T Oct. 2008 Pittsburgh
4. Summary
1. DFT and homogenization schemes can be used to predict with reasonable accuracy elastic properties of polycrystalline metals
2. Optimal elastic properties of bcc MgLi alloys are observed around 70 at. % Mg
3. B/G for the optimal bcc Mg-Li alloys is in the brittle/ductile transition region
4. BCC MgLi has a better E/r than AlMg and a comparable E/r to Al-Li
BCC MgLi HAS POTENTIAL AS AN ULTRA-LIGHT WEIGHT STRUCTURAL ALLOY
MS&T Oct. 2008 Pittsburgh
ConclusionsConclusions
+ Understanding trends (thermodynamics, mechanics)
+ Direct use of homogenization theory (elastic)
+ Extract engineering quantities for a rough but quick estimation
+ Get quantities that you cannot get elsewhere
- 0 K
- supercell size
- long calculation times