Application of Crystal Plasticity to Fatigue Damage Modeling for...

Application of Crystal Plasticity to FatigueDamage Modeling for Al 7075-T651

David Littlewood and Antoinette [email protected]

Mechanical, Aerospace, and Nuclear Engineering

Rensselaer Polytechnic Institute

Collaborators: Tony Ingraffea (Cornell), Tony Rollett (Carnegie Mellon),

Hasso Weiland (Alcoa), Mark Horstemeyer (Mississippi State),

Bob Wei and Gary Harlow (Lehigh)

Sponsor: DARPA SIPS program (Northrop-Grumman)

WCCM 2006 – slide 1/19

Outline

F Motivation

F Methodology

F Constitutive model

F Finite element formulation

F Metrics for damage accumulation

F Implementation

F Results

F Calibration results

F Example: 45-grain polycrystal

F Future work


Motivation

F Fatigue cracks are observed to initiate primarily atcracked particles in Al 7075-T651

F The length of fatigue cracks is not observed to be afunction of particle size

F Approach ⇒ focus on microstructure

[WEI AND HARLOW]


Multiscale Nature of Problem

[WEI AND HARLOW, WEILAND, WIEZOREK]


Collaborative Approach

Orowan Looping

Cornell, RPI

RPI

ConstitutiveCrystal

Model

UnderlyingPhenomena

ExperimentalObservations

Lehigh, AlcoaCMU, MSU, NG

PolycrystalModel


Methodology

Constitutive Model

Deformation 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

[Kocks (1966), Hart (1972), Schmitt, Lipinski,

Berveiller (1997), Han, Wagoner and Barlat (2004)]


Methodology

Slip Model

Plastic velocity gradient pL =∑12

α=1 γα Pα

Schmid tensor Pα = sα⊗ mα

Slip rate γα = γoτα

gα

∣∣∣ τα

gα

∣∣∣1

m−1

Hardness evolution gα = Go

(go−gα

gs−go

)Hαβ

Projection factor Hαβ =∑

β 2∣∣∣Sα

ij Sβij

∣∣∣∣∣γβ

∣∣


Methodology

Finite Element Formulation

Governing equations:(σ

′

ij + p δij

)

,j= 0, 1

3σii − 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


Methodology

Finite Element Formulation

Linearized 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

)


Methodology

Transition from Damage Accumulation to Fatigue Cracking

F Trigger fatigue cracking with plastic damage metrics

F Fatigue cracks inserted in model (Cornell)

F Metrics for damage accumulation

F Total accumulated plastic slip over all slip systems

F Maximum accumulated plastic slip on a single slip

system

F Maximum accumulated plastic slip on a slip plane

WCCM 2006 – slide 10/19

Implementation

F Software developed in C++

F Developed within Cornell FEM framework

F Utilized MPICH for parallel processing

F PETSc used for solving global system of equations

MPICH

Crystal PlasticityModel

RPI

FemLib (Cornell)

RPIDriver

DriverCornell

PETSc

WCCM 2006 – slide 11/19

Implementation

F Running on large-scale systemsF SCOREC Medusa RedHat Linux cluster

? 32 nodes, dual Xeon 2.0 GHz processors, 2 GB RAM

F Cornell Theory Center Windows cluster? 170 nodes, dual Xeon 3.6 GHz processors, 4 GB RAM

F SDSC DataStar AIX Unix system (p655 partition)? 272 nodes, eight 1.5 GHz processors, 16 GB RAM

F Performance analysis [Scott Owens]

WCCM 2006 – slide 12/19

Calibration Results

Calibrated Material Parametersm 0.005 γo 1 s

−1

go 220 MPa µ 28.3 GPa

gs 250 MPa λ 60.9 GPa

Go 120 MPa η 5.1 GPa

WCCM 2006 – slide 13/19

Polycrystal with Embedded Particle

ORIENTATION SET #1 ORIENTATION SET #2

WCCM 2006 – slide 14/19

Results

TOTAL ACCUMULATED SLIP, TIME = 2.0 SEC.

WCCM 2006 – slide 15/19

Results


WCCM 2006 – slide 15/19

Results

MAX ACCUMULATED SLIP ON SINGLE SLIP SYSTEM, TIME = 2.0 SEC.

WCCM 2006 – slide 16/19

Results


WCCM 2006 – slide 16/19

Results

MAX ACCUMULATED SLIP ON SINGLE SLIP PLANE, TIME = 2.0 SEC.

WCCM 2006 – slide 17/19

Results


WCCM 2006 – slide 17/19

Future Work

F Project results into high-cycle regime

F Move between different time scales

F Fatigue crack insertion

F Dictated by plastic damage metrics

F FEM model altered to include crack (Cornell)

F Fatigue crack propagation

F Direction of propagation influenced by microstructure

F Particle-free zones

F Material properties altered near grain boundaries

WCCM 2006 – slide 18/19

Questions?

Application of Crystal Plasticity to FatigueDamage Modeling for Al 7075-T651

David Littlewood and Antoinette Maniatty

[email protected]

WCCM 2006 – slide 19/19

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