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Multi-Scale Crystal Plasticity FEM Approach To Modeling Nickel-
based Superalloys
Somnath Ghosh (M. G. Callas Professor) Shahriyar Keshavarz (Post-Doctoral Researcher)
Collaborators: M.J. Mills, Y. Wang (Ohio State U)
Departments of Civil & Mechanical Engineering
Johns Hopkins University.
Sponsors: GE Aviation, AFOSR AIAA SciTech 2014
January 15, 2014 National Harbor, MD
Commercial use of Ni-Base Superalloys as Turbine Disks and Blades
Introduction Sub-Grain Scale Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
𝛾-phase face-centered crystal (fcc) lattice γ’-phase is an ordered L12 intermetallic containing Ni3Al
Turbine Blade
Single Crystals Higher γ’ volume fraction, Higher temperature, lower stress
Polycrystals Lower γ’ volume fraction, Lower temperature, higher stress
Turbine Disk
Deformation Mechanisms in Nickel-Based Superalloys
1
R.R. Unocic et al. Acta Materialia 59 (2011)
Secondary 𝜸𝜸 phase
𝜸 −phase
APB-shearing
Micro-twinning
Introduction Sub-Grain Scale Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Objective: Develop a Physically Motivated Hierarchical Multi-Scale Framework
Single Grain Level Model
Deformation Mechanisms: γ−γ’ morphology dependent slip and intra-granular micro-twinning
• Homogenized crystal plasticity model accounting for sub-grain phase-morphology •Micro-twinning model
Sub-Grain Level Model
Micro-structure FE model Deformation Mechanisms: • Dislocation density based
crystal plasticity • APB shearing in 𝜸𝜸phase
Polycrystalline Model
Strain
Stre
ss(P
a)
0.01 0.02 0.03 0.040
1E+08
2E+08
3E+08
4E+08
5E+08
6E+08
7E+08
8E+08
9E+08
Homogenization
Macroscopic Experiments
Introduction Sub-Grain Scale Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Sub-Grain Model With 𝜸 − 𝜸’ Phase Morphology
Polycrystalline level
Grain scale Sub-Grain scale
Homogenized single crystal level
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Shahriyar, Ghosh, Acta Materialia, 2013 Shahriyar, Ghosh, IJSS (submitted)
Sub-Grain Crystal Plasticity FEM
•Dislocation density-based crystal plasticity model
•Explicit representation of 𝜸 and 𝜸’ phase morphology
•APB shearing in the 𝜸’ phase
•CP parameters calibrated from single crystal experiments
•Model validated with single crystal creep test results
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Sub-Grain Model with γ−γ’ Phase Morphology
γ-γ’ micro-structure
γ phase fcc structure
Secondary γ’ phase ordered L12 structure
tertiary γ’ phase ordered L12 structure
Morphology of γ’ phase: o Different channel width, shape and volume fraction
5
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Operative Mechanisms: o Dislocation glide in the γ-phase matrix causing plastic deformation o Dislocation generation and annihilation in the γ-phase o No initial dislocations in the γ’ phase o APB shearing of γ’ phase by matrix dislocations
• Dislocations in 12 slip systems (FCC phase)
• Onset of plastic deformation after a threshold stress is reached
• Velocity of dislocations depends on applied resolved shear stress τα in the slip systems and the slip system resistances (τpass and τcut )
0 when passvα α ατ τ> > Ma, Roters, Acta Mater. (2006)
Crystal Plasticity Model for 𝜸-Phase Matrix
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Slip System Flow Law (from Orowan equation)
Strain hardening is due to resistance of forest dislocation (other planes crossing the plane) and parallel (in the same plane) dislocations
Crystal Plasticity Model for 𝜸-Phase Matrix
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
( ) ( ) ( ) ( )1
cos , cos , cos , cos ,α αβ β α β β α β β α β β α β
β
ρ χ ρ ρ ρ ρ=
= + + + ∑N
F SSD GNDs GNDet GNDenn t n d n t n n
( ) ( ) ( ) ( )1
sin , sin , sin , sin ,α αβ β α β β α β β α β β α β
β
ρ χ ρ ρ ρ ρ=
= + + + ∑N
P SSD GNDs GNDet GNDenn t n d n t n n
Forest and parallel dislocation densities
Dislocation Evolution Relations
Geometrically Necessary Dislocations (screw, edge and normal components)
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Statistically Stored Dislocations (lock formation, dipole formation, athermal annihilation and thermal annihilation)
Crystal Plasticity Model for γ’-Phase
• Initial density of dislocation is zero in γ’ (ordered) phase • 12 slip systems • Matrix dislocations form super-dislocations at the interface of matrix and precipitates; enter the ordered phase through anti-phase boundary (APB ) shearing
c m cα ατ τ ρ ρ> >
Constitutive Model for γ’ phase
( )
m m( )
( )exp 1 sgn
H
H
c
pass c
B cut
bv
QvK T
α α α α
α α αα α
α
γ ρ ρ ρ
τ τ τ τλν τ
τ
= −
− − = − −
m F Pwhere cTα α αρ ρ ρ=
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
• APB shearing occurs when the resolved shear stress along the slip system and the dislocation density at the interface exceed critical values
S. Keshavarz and S. Ghosh, Acta Materialia (2013)
Comparing Results Based on Computational Model with Experiments
(750oC, 770 MPa) of a single crystal with 70% volume fraction of secondary γ’ in the shape of cubes with an edge length of 0.45
Time, h
Stra
in
0 10 20 30 40 50 60 70 800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
770-experimental1770-computational
Experimental data are based on:N. MATAN, D.C. COX, P. CARTER, M.A. RIST,C.M.F. RAE and R.C. REED,Acta ma. 47, 1999.
Creep Response
Validation of Sub-Grain Computational Model with Experiments
Strain
Stre
ss(M
Pa)
0 0.01 0.02 0.03 0.04 0.050
100
200
300
400
500
600
700
800
900
1000
1100
Experimental resultsComputational results
Fleury et al. Computational Materials Science 7 (1996)
Constant Strain Rate
(750oC, constant strain rate 0.01% ) of a single crystal with 70% volume fraction of secondary γ’ in the shape of cubes with an edge length of 0.45
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Effect of the Channel Width
Strain
Stre
ss(M
Pa)
0 0.01 0.02 0.03 0.04 0.050
150
300
450
600
750
900
1050 Symmetric microstructure with one channel widthAymmetric microstructure with two channel widths
Simulations have been done under 750oC, constant strain rate 0.005% of a single crystal with 35% volume fraction of secondary γ’ in the shape of cubes with RVE size of
Comparison between symmetric and asymmetric precipitates
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Activation of APB Shearing on Mobile Dislocation (Cuboidal γ’)
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Effect of Morphology and Mechanisms on Stress-Strain Response
Without APB-shearing
Without precipitates
With APB shearing
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Activation of APB Shearing on Mobile Dislocation (Spheroidal γ’)
Effect of Morphological Variables γ’ Precipitate Shape
11 tan ( )n n−=Shape factor of precipitates:
X
X
X X
Z
n=4.27 9.55 ∞
1n n nx y z
a b c + + =
n=1.5 2.0 2.79
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
2. Volume fraction of precipitates
Precipitate Volume Total Volume
With increasing volume fraction of precipitates, the yield stress increases The increase becomes significant at larger volume fractions
Effect of Morphological Variables γ’ Precipitate Volume Fraction
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
clChannel width between precipitates:
With increasing channel width, the yield stress decreases With increasing channel width, the hardening response changes
Effect of Morphological Variables γ-Channel Width
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Effect of Channel Width on Stress-Strain Response
Strain
Stre
ss(M
Pa)
0 0.01 0.02 0.03 0.04 0.050
200
400
600
800
1000
1200
3978157394
Channel width (nm)
Strain
Stre
ss(M
Pa)
0 0.01 0.02 0.03 0.04 0.05 0.060
200
400
600
800
1000
1200
2254509002250
Channel width (nm)
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Grain Level Crystal Plasticity and Micro-Twinning Model
Sub-Grain scale
Homogenized polycrystalline level Homogenized single crystal level
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Grain-Level Homogenized Crystal Plasticity Model
Thermally activated theory of plastic law rate on a slip system:
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Yield point phenomena due to rapid dislocation multiplication.
Grain-Level Homogenized Crystal Plasticity Model
Thermally activated theory of plastic law rate on a slip system:
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Evolution of thermal and athermal shear resistances
Validation of Grain Scale Constitutive Parameters with Experiments
Simulation have been compared with experimental data taken from creep tests (750oC, 770 MPa) of a single crystal with 70% volume fraction of secondary γ’ in the shape of cubes with an edge length of 0.45. Knowles, et. Al. 2002, Matan et. al. 1999.
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
11 tan ( )n n−=
1. Shape factor of precipitates:
XX
Z
1n n nx y z
a b c + + =
n=1.5 2.0 4.27 ∞
2. Channel width between precipitates lc:
3. Volume fraction of precipitates:
Homogenized Parameters
Morphological and Homogenized Parameters
* * 1
1
* * 1
1. ( , , , )
2. ( , , , )
3. ( , , , )
p c
sat sat p c
p c
s s n v l
s s n v l
n v l
α α
α α
α α
γ
γ
γ γ γ
=
=
=
pv
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Homogenization from Stress-Strain Response: Calibration of Parameters
Hill-Mandel Principle Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Variation of Initial Thermal Shear Resistance with Channel Width
Variation of Initial Thermal Shear Resistance with Channel Width for Different Volume Fraction and n=4.27
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Homogenized Parameters Obtained from Sub-Grain Scale Model
(I)
1 1* 0 1 1 1
90 89 53 0.1( , , ) 136 559 99 1039 p p
p c p pc
v n v ns n l v n v n
lα ν
− + + −= − + − + +
(II)
1 11 1 1
3599 5008 363 0.21( , , ) 6680 8905 1648 3185 p p
sat p c p pc
v n v ns n l v n v n
lα ν
− + + −= − − + +
(III) 1 1 1
1 1
( , , ) 19847 12768 23120
4080 7500 33 2700 65p c p c p p c
c p c
k n l v n l v n v ln l v n l
ν∗ = + − +
− + − +
(IV)
1 11 1 1
176.5 281.2 2.44 0.14( , , ) 221.4 327.6 31.5 5.5 p p
p c p pc
v n v nk n l v n v n
lν
− + − += − + + +
S. Keshavarz and S. Ghosh, Acta Materialia
(2013)
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Validation of the Homogenized Model
Simulations have been done under (750oC, constant strain rate 0.0001 1/s ) of a single crystal with different shape volume fraction and channel width of secondary γ’ for two scales: 1) SG-RVE with explicit precipitate
expression 2) AE-CP with implicit precipitate
expression(homogenized functional forms)
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Validation of Homogenized Grain Scale with Experiments
Simulation have been compared with experimental data taken from creep tests 750oC, 770 Mpa in [001] direction and 800oC, 675 Mpa in [111] direction of a single crystal with 70% volume fraction of secondary γ’ in the shape of cubes with an edge length of 0.45.
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Grain Level Micro Twinning
• Twin Nucleation Criterion
• Constitutive Model For Micro Twinning
• Asymmetry in Tension And Compression
•Model Validation with Single Crystal Creep Test Data
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Criterion For Twin Nucleation Dissociation of Leading and Trailing Partials
Critical configuration for the leading partial to pass
through the channel
Critical configuration for the trailing partial to pass
through the channel
Decorrelation of the leading and trailing partials
R.R. Unocic et al. Acta Materialia 59 (2011)
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Based on the state of dissociation of the leading and trailing partials on a slip system. Condition for dissociation of a full dislocation a/2<110> is given as a function of the magnitude and orientation of the in-plane shear stress.
,inplane inplanelead trailτ τ τ τ> <
,inplane inplanelead trailτ τ τ τ> <
;in plane leadcr
in plane trailcr
τ τ
τ τ
−
−
>
<
2 ( ) cos( ) cos( )cos( ) cos( )
leadcr f l t
l t
bαµτ τ θ θθ θ λ
= + ∀ ≤+
Critical stress for leading and trailing partials:
Criterion For Twin Nucleation Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
Constitutive Model for Twin Evolution
3
2pt
efftp
fb
τ τΓ
= −
0 1 2 3 4 5x 106
0
0.5
1
1.5
2
2.5
3 x 10-7
time (sec)
stra
in ra
te (1
/sec
)
predictionexp. (Viswanathan et al. 2005)
Twin shear strain accumulation: Precipitate shearing and subsequent re-ordering is the predecessor to the movement of partials causing plastic slip
Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level
γ ′precipitate shearing and subsequent re-ordering is the predecessor to the movement of partials causing plastic slip.
Shahriyar and Ghosh Acta Mat. 2013
( ) ( )exp( )pt tt ttt KtΓ = Γ − Γ − + Γ
Energy drop decreases exponentially with time from pseudo-twin energy to true twin energy
Summary of CPFEM Developments
1. Sub-Grain Scale •Dislocation density based crystal plasticity model • Explicit representation of γ and γ’ phases with different morphology • APB shearing model • Model parameters calibrated from single crystal constant strain rate experiments • Model validated with single crystal creep test results
2. Grain Scale-Single Crystal Level • Dislocation density based homogenized crystal plasticity model • Parametric functional dependent on γ and γ’ phase morphologies • Implementation of the micro-twin nucleation criterion • New constitutive model for microtwinning in the presence of slip • Model validated with single crystal creep test data with tension-compression asymmetry
3. Grain Scale-Poly Crystal Level • Generate 3D Microstructures from 2D Images and • Incorporate Grain Based Constitutive Laws for Homogenized APB Shearing and Microtwinning • Prediction of experimental data for the polycrystalline microstructure through the homogenized crystal plasticity and micro-twinning model