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Multiscale Modeling of Crystal Plasticity in Al 7075-T651 David Littlewood and Antoinette Maniatty Mechanical, Aerospace, and Nuclear Engineering Rensselaer Polytechnic Institute Troy, New York USA Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.1/17
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Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Apr 20, 2022

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Page 1: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Multiscale Modeling of CrystalPlasticity in Al 7075-T651

David Littlewood and Antoinette Maniatty

Mechanical, Aerospace, and Nuclear Engineering

Rensselaer Polytechnic Institute

Troy, New York USA

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.1/17

Page 2: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

OutlineMotivation

Methodology

Constitutive model

Finite element formulation

Implementation

Results

Calibration results

Model behavior

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MotivationPredict failure of Al 7075-T651 under spectrum loading

Fatigue crack initiation at large particles (e.g. Al7Cu2Fe)

Determine which large particles will produce cracks

Focus on crystallography

Key phenomena that must be captured:

Material hardening

Geometric effects (grain structure)

Texture effects (orientation)

Damage accumulation (irreversible slip)

Particle effects

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MethodologyModel developed within a collaborative environment:

Underlying

Phenomena

(Nature)

Crystal

Constitutive Model

(RPI)

Polycrystal

FEM Model

(CMU, CU, RPI)

Experimental

Observation

(Small Scale − CMU, Alcoa)

(Large Scale − MSU, NG)

Multiscale modeling approach:

Macro-scale (continuum) FEM models provide boundaryconditions for grain-scale RVE modeling

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.4/17

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Constitutive ModelDeformation gradient F = eF pF

Green strain tensor eE = 12

(eFT eF − I

)

Hyperelastic potential ψ = 12

eE : C : e

E

Second Piola-Kirchoff stress S = C : eE

Anisotropic elasticity Cijkn = Cjikn = Cijnk = Cknij

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Plastic Slip Model

Power law determines slip rates γα = γoτα

∣∣∣τα

∣∣∣1

m−1

Hardness evolution dominated by Orowan looping

Strong self hardening

gα = Go

(gs−gα

gs−go

)∑β 2

∣∣∣Sαij Sβ

ij

∣∣∣∣∣γβ

∣∣

Evolving slip-system activity

OROWAN LOOPING

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.6/17

Page 7: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Finite Element Formulation3D formulation with additional pressure variable for stability

Governing equations:(σ

ij + p δij

),j

= 0, 13σii − p = 0

Weak forms (total Lagrangian):∫

Ωo

(σh

ij + ph δij

)ψα,KF−1

KjJdΩo

︸ ︷︷ ︸f int

iα(u,p)

∂Ω2o

tiψαdA

dAodΓo

︸ ︷︷ ︸fext

= 0

Ωo

1

K

(1

3σh

ii − ph

)ψρJdΩo

︸ ︷︷ ︸hρ(u,p)

= 0

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.7/17

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Finite Element FormulationLinearized equations:

Kriαjβ ∆ujβ + Gr

iαϕ ∆pϕ = f extiα − f int

iα (ur, pr)

Hrρjβ ∆ujβ + Mr

ρϕ ∆pϕ = 0 − hρ (ur, pr)

Discontinuous pressure field allows for a ∆p solution on theelement level:

∆pϕ = −Mr−1

ϕρ

(hρ (ur, pr) + Hr

ρjβ ∆ujβ

)

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.8/17

Page 9: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Integration RoutinePlastic velocity gradient in terms of slip rates

pL =Nss∑

α=1

γα(pF) Pα

Plastic velocity gradient by integration

pL = pF pF−1

Solve non-linear residual equation for pF

R =

Nss∑

α=1

γα(pF) Pα

pF pF−1

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Page 10: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

ImplementationCrystal plasticity model implemented in C++

Implemented consistent tangent formulation

Update state and tangent routines incorporated intoexisting library of finite-element routines

Finite-element driver implemented at RPI for calibrationand testing

Utilizes MPICH for parallel processing

PETSc software package used for solving globalsystem of equations

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Computational Resources

Code development PowerMac G5

Dual 2.8 GHz processors, 8 GB RAM

Small- to mid-sized models SCOREC

Linux cluster

32 nodes: single Xeon 2.0 GHz processor, 2 GB RAM

Large-scale models Cornell Theory Center

Windows cluster

170 nodes: dual Xeon 3.6 GHz processors, 4 GB RAM

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Example Runs

machine processors d.o.f. steps run time

G5 2 20,577 100 1.1 h

SCOREC 6 20,577 100 0.8 h

SCOREC 8 46,875 100 1.2 h

CTC 32 3,594,558 102 23 h

CTC 64 6,516,492 128 46 h

CTC 128 6,516,492 128 24 h

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Model CalibrationCalibration performed against monotonic and cyclic

experimental data (MSU, NG)

SINGLE TENSION/COMPRESSION CYCLE

Model Parameters

m 0.005

go 220 MPa

gs 350 MPa

Go 120 MPa

γo 1 s−1

µ 28.3 GPa

λ 60.9 GPa

η 5.1 GPa

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Model BehaviorHardness evolution under cyclic loading

TENSION/COMPRESSION CYCLES

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Model BehaviorEffects of grain orientation and particle inclusions

SINGLE-GRAIN MODEL WITH PARTICLE (CU)

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Model BehaviorEffect of crystal orientation on plastic slip accumulation

(100) POLE FIGURE FOR AL-7075 LOADED IN THE ROLLING DIRECTION

MAXIMUM PLASTIC SLIP ON SINGLE SLIP SYSTEM

0.5% STRAIN

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Model BehaviorEffect of crystal orientation on plastic slip accumulation

(100) POLE FIGURE FOR AL-7075 LOADED IN THE ROLLING DIRECTION

MAXIMUM PLASTIC SLIP ON SINGLE SLIP SYSTEM

1.0% STRAIN

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.16/17

Page 18: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Model BehaviorEffect of crystal orientation on plastic slip accumulation

(100) POLE FIGURE FOR AL-7075 LOADED IN THE ROLLING DIRECTION

MAXIMUM TOTAL PLASTIC SLIP

0.5% STRAIN

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.16/17

Page 19: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Model BehaviorEffect of crystal orientation on plastic slip accumulation

(100) POLE FIGURE FOR AL-7075 LOADED IN THE ROLLING DIRECTION

MAXIMUM TOTAL PLASTIC SLIP

1.0% STRAIN

Multiscale Modeling of Crystal Plasticity in Al 7075-T651 – p.16/17

Page 20: Multiscale Modeling of Crystal Plasticity in Al 7075-T651

Acknowledgements

DARPA

H. Weiland Alcoa

A.D. Rollett Carnegie Mellon

J. Papazian Northrop-Grumman

A. Ingraffea, P. Wawrzynek, G. Heber, A. Liu Cornell

M. Horstemeyer, Y. Xue, B. Jordan Mississippi State

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