Multiscale Modeling of Crystal Plasticity in Al 7075-T651
Transcript of 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
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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
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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
gα
∣∣∣τα
gα
∣∣∣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
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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
iα
= 0
∫
Ωo
1
K
(1
3σh
ii − ph
)ψρJdΩo
︸ ︷︷ ︸hρ(u,p)
= 0
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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
)
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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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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
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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 TOTAL PLASTIC SLIP
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 TOTAL PLASTIC SLIP
1.0% STRAIN
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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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