multi-scale crystal plasticity fem approach to modeling ... documents/2014 icme... · multi-scale...

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

http://www.jhu.edu/

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

http://www.jhu.edu/

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


http://www.jhu.edu/

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


http://www.jhu.edu/

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)

http://www.jhu.edu/

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


http://www.jhu.edu/

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


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

http://www.jhu.edu/

• 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


Slip System Flow Law (from Orowan equation)

http://www.jhu.edu/

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


( ) ( ) ( ) ( )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

http://www.jhu.edu/

Dislocation Evolution Relations

Geometrically Necessary Dislocations (screw, edge and normal components)


Statistically Stored Dislocations (lock formation, dipole formation, athermal annihilation and thermal annihilation)

http://www.jhu.edu/

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=


• 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)

http://www.jhu.edu/

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


http://www.jhu.edu/

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


http://www.jhu.edu/

Activation of APB Shearing on Mobile Dislocation (Cuboidal γ’)


http://www.jhu.edu/

Effect of Morphology and Mechanisms on Stress-Strain Response

Without APB-shearing

Without precipitates

With APB shearing


http://www.jhu.edu/


Activation of APB Shearing on Mobile Dislocation (Spheroidal γ’)

http://www.jhu.edu/

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


http://www.jhu.edu/

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


http://www.jhu.edu/

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


http://www.jhu.edu/

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)


http://www.jhu.edu/

Grain Level Crystal Plasticity and Micro-Twinning Model

Sub-Grain scale

Homogenized polycrystalline level Homogenized single crystal level


http://www.jhu.edu/

Grain-Level Homogenized Crystal Plasticity Model

Thermally activated theory of plastic law rate on a slip system:


Yield point phenomena due to rapid dislocation multiplication.

http://www.jhu.edu/

Grain-Level Homogenized Crystal Plasticity Model

Thermally activated theory of plastic law rate on a slip system:


Evolution of thermal and athermal shear resistances

http://www.jhu.edu/

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.


http://www.jhu.edu/

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


http://www.jhu.edu/

Homogenization from Stress-Strain Response: Calibration of Parameters

Hill-Mandel Principle Introduction Sub-Grain Scale Model Single Crystal Level Homogenization Micro Twinning Polycrystalline Level

http://www.jhu.edu/

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


http://www.jhu.edu/

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)


http://www.jhu.edu/

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)


http://www.jhu.edu/

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.


http://www.jhu.edu/

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


http://www.jhu.edu/

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)


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

−

−

>

<

http://www.jhu.edu/

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

http://www.jhu.edu/

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


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

http://www.jhu.edu/

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

http://www.jhu.edu/

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