Department of Mechanical Science and Engineering

University of Illinois at Urbana-Champaign

www.mechse.uiuc.edu1

Recent Advances in Modeling Fatigue in Metals

Symposium in honor of C. TomeTMS, San Diego, February 28, 2011

Work Supported by Rolls Royce, National Science Foundation, DMR, Metals

H. Sehitoglu, M. Sangid,T. Ezaz, H.J.MaierUniversity of Illinois, USA, University of Paderborn, Germany

OutlineModeling of Fatigue• Analysis of Grain Boundaries- Energy Barriers

for G.B. Slip Transmission And G.B. Slip Nucleation

• Energy Formulation for Crack Initiation via Persistent Slip Bands, Life results

Fatigue Crack InitiationExperimentally observed mechanism: transgranular facets forming from persistent slip bands (PSBs) across GBs

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

Atomic SimulationsUse Molecular Dynamics Code, LAMMPs • Construct GBs from crystal lattices (axis/angle pairs)• Ni with Foiles-Hoyt EAM Potential: FCC Structure• 3D Periodic Boundary Conditions• Atoms ‘relax’ to determine grain boundary energy

GB ECSL

GB N

MEPerfect

FCC

Area

Atomic Simulations of Prevalent Tilt & Twist GBs

<001>

( <111>θ

<110>

θ (

0

200

400

600

800

1000

1200

1400

1600

0 30 60 90 120 150 180

Rotation Angle (degrees)

Gra

in B

ou

nd

ary

En

erg

y (m

J/m

2 ) <110> Tilt <111> Twist <001> Tilt

Perfect FCC

Perfect FCCΣ3

Σ9

Σ11

Σ17

Σ19Σ5 Σ5

Σ3 Σ3

Σ7Σ13

Σ21

Σ7Σ13

Σ21

Σ7Σ13

Σ21

Mechanical Behavior of GBs: Slip Transmission

A Σ3 (Twin) GB

Y [111]

Z [011]

X [211]

a/2 [011] t (100)t

a/6 [211](111)

Stair RodLomerIncident

a

6211 a

2011 T

a

3100

Sangid MD, Ezaz T, Sehitoglu H, Robertson IM, “Energy of slip transmission and nucleation at grain boundaries,” Acta Materialia 59 283-296 (2011).

Contour Plot of Energy Barriers

0

50

100

150

200

250

300

350

400

0 0.5 1 1.5 2Reaction Coordinate: Uz/<112>/6

Sta

ck

ing

Fa

ult

En

erg

y (m

J/m

2)

Static GSFE Curve Control Box Method

DFT Data from Siegel, 2005 Experimental Data

GB Energy Barriers to Slip• Monitor the energy of the atoms within a

control box at the GB

• Obtain the energy barrier for slip to penetrate the GB

• Validated values of slip in a perfect FCC lattice by comparing with GSFE curve

volume

EEE

n

istatic

iload

Error ~ 6%

Measured Energy Barrier for Slip Transmission for various GBs

Validation of MD Results of Slip Transmission

Criterion for Slip Transmission• Geometrical condition

– Minimizes angle between lines of intersection of slip planes with GB, maximize M:

• Resolved shear stress condition– After predicting active slip

plane from GC, choose direction based on max resolved shear stress

• Residual grain boundary dislocation condition

– Minimize Burgers vector of residual dislocation (difference in b of incoming and outgoing ):┴

Lee, Robertson, and Birnbaum, 1989

0.0E+00

5.0E+11

1.0E+12

1.5E+12

2.0E+12

2.5E+12

0 200 400 600 800 1000 1200 1400

Static GB Energy (mJ/m2)

En

erg

y B

arri

er f

or

Dis

loca

tio

n -

GB

Inte

ract

ion

(m

J/m

3 )

Σ7

Σ21

Σ17

Σ9Σ19

Σ5

Σ3

Σ11

Σ13

Perfect FCC

Energy Barriers for Slip Transmission through GB

6.013108.2

GBStatic

onTransmissiBarrier EE

Twin boundary has inherently high energy barriers for slip transmission

As the applied load increases, slip transmits past the twin boundary.

Slip is initially impeded by the twin resulting in a dislocation pile-up.

Ezaz T, Sangid MD, Sehitoglu H, “Energy barriers associated with slip-twin interactions,” Philosophical Magazine, (2011).

Coherent twin boundary• Stable defect structure• Low static GB energy• High energy barrier for

slip transmissionNote:• Applied loading is normal to GB• No applied shear stress on GB• General twin-slip interaction

energies please refer to:

DOI: 10.1080/14786435.2010.541166

Measured Energy Barrier for Slip Nucleation from various GBs

0.0E+00

2.0E+11

4.0E+11

6.0E+11

8.0E+11

1.0E+12

1.2E+12

1.4E+12

1.6E+12

1.8E+12

2.0E+12

0 200 400 600 800 1000 1200 1400

Static GB Energy (mJ/m2)

En

erg

y to

Nu

clea

te a

Dis

loca

tio

n (

mJ/

m3) Σ7

Σ21Σ13

Σ17

Σ9

Σ19

Σ5

Energy Barriers for Slip Nucleation from GBPlease note: the Σ1,3,&11 GBs have a simple dislocation structure and stable configurations. Hence dislocations were nucleated in the matrix material during the simulation, preventing the energy barrier to be measured

Fatigue Crack InitiationExperimentally observed mechanism: transgranular facets forming from persistent slip bands (PSBs) across GBs

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

PSB Modeling and Energy Contributions• Model the energy of a persistent slip band through a physics

based approach• Create an energy balance evolving with increasing loading

cycles, which addresses all the PSB energy contributions:– Stress-field which must be overcome to have slip within the PSB

• Applied Strain• Dislocation-dislocation interaction within the PSB (work-hardening)• Internal stress-field from dislocations

– PSB interaction with GBs, particularly dislocations:• Piling-up at GBs• Nucleating from GBs and agglomerating within the PSB• Transmission through the GB

– Formation of the PSB, dislocations shearing the γ matrix and γ’ precipitates

• Crack initiates when PSB reaches minimum energy wrt to plastic deformation, i.e. dislocation motion

Energy Formulation for a PSB

Extrusion Formation at GBs

Applied Work

Conti

nuum

MD

Dislocation Pile-ups

Shearing of γ’ PrecipitatesDislocation Nucleation at GBs

where:

σ ≡ Applied stress

h ≡ Width of PSB

d ≡ Dislocation spacing

ρ ≡ PSB dislocation density

N ≡ Number of cycles

Monitor a PSB and when it reaches a stable point, the material fails!

Work Hardening in Bands

Shearing of γ Matrix

Energy Eapp (,m,L,N) Ehard (,L,N) E pile up

disl (h,d,L,N)

Enucdisl (m,,h,L,L',N) E extrusion

slip GB (m,,h,L,L',N) EAPB (L, dist ,N) E SF (L, dist ,N)

Each term is expressed in terms of a slip increment, ∂X

m ≡ Schmid factor

L ≡ Grain size

Σ ≡ Characteristic of GB

γ’ ≡ Distribution of precipitate

L’ ≡ Grain size of neighboring grain

Exp

func

tions

Mic

rost

ruct

ure

Inpu

ts

Output

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

nlayers = number of dislocation layers in the PSB related to width of PSB, normalized by annihilation distance

Energy associated with overcoming stress field within the PSB for motion of glissile dislocations

dis A hwhere the total stress is given by:

Total stress

Pile-up of dislocations

Applied stress

Lattice resistance

E =rbLnlayers∂X

L = grain size

∂X = increment of slip

b = Burgers vector

where:

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

PSB-GB Interaction EnergyAtomistic Based Formulation:• Dislocations nucleate from the GB and localize

in slip bands

– The number of dislocations that are emitted from the GB and aggregate within the PSB is given by:

• PSBs form extrusions at grain boundaries in polycrystalline material

• Leverage energy barriers for slip transmission and nucleation at a GB from atomic simulations (previously shown)

Eextrusion

slip−GB = ∂Xi ⋅EMD−slip−GBndis

penrb

i∑ h

Enuc

disl = ∂Xi ⋅EMD−nuc−GB −o( )

rbhL2

i∑

o hL

Sangid MD, Maier HJ, Sehitoglu H, “The role of grain boundaries in fatigue crack initiation – an energy approach,” Accepted to International Journal of Plasticity (In Press - 2011).

Energy due to shearing γ matrix and γ’ precipitates

Shearing of γ’ precipitates O O’M M’R

Atomistic Based Formulation:

Stacking Fault Energy + Anti-Phase Boundary Energy

where f is the area fraction of γ’, fU720 ~ 0.20

EAPB + E−SF = f APB dLo

d

∫ + 1− f( ) SF dLo

d

∫( )nefflayers∂X

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

PSB Energy Balance

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

Sangid MD, Ezaz T, Sehitoglu H, Robertson IM, “Energy of slip transmission and nucleation at grain boundaries,” Acta Materialia 59 283-296 (2011).

Criterion for Crack Initiation• Create an expression for energy (as energy components were

previously shown) • Increment the number of loading cycles and update the energy

expression as variables evolve:

• Criterion for crack initiation– Energy of PSB reaches a stable value– Minimize energy to check for stability of PSB:

where Xi represents the position of the glissile (mobile) dislocations in our energy expression

0

iX

E

layerseff

layerspendis

A nnndh ,,,,,,

Sangid MD, Maier HJ, Sehitoglu H, “A physically-based model for prediction of crack initiation from persistent slip bands in polycrystals,” Acta Materialia 59 328-341 (2011).

More on PSB-GB Interaction

TEM pictures from: Zhang & Wang, Acta Mat 51 (2003).

θ < 15˚

θ > 15˚

PSBs transmit through low-angle GBs, θ < 15˚

High-angle GBs, θ > 15˚, impede dislocations, resulting in pile-ups, stress concentration, local increase in energy, and ultimately crack initiation

Clustering of grains

Grain cluster (multiple grains connected by LAGBs)

Failure occurs due to PSB-GB interaction

Single large grain

or

Fatigue Scatter Results

0.6

0.7

0.8

0.9

1

1.1

1.2

100 1000 10000 100000 1000000Cycles to Initiation

No

rmal

ized

Ap

plie

d S

trai

n R

ang

e, %

Model - Simulated SpecimensModel - AverageU720 Experimental DataU720 Data - Average

1000 simulated specimens vs. 84 experimental results

Each simulated specimen takes <30 seconds to construct its microstructure and predict fatigue life for a series of strain ranges

Sangid MD, Maier HJ, Sehitoglu H, “An energy-based microstructure model to account for fatigue scatter in polycrystals,” Journal of the Mechanics and Physics of Solids (In Press - 2011).

Conclusions• Atomistic Simulations

– Quantified the strengthening mechanisms of slip transmission and nucleation from GBs

– Inverse correlation between energy barrier and static interfacial energy

• Lower GB energy results in a stronger barrier.

• Fatigue modeling– Introduced methodology to model persistent slip bands energetics, in

order to predict fatigue life.

• Prediction of fatigue life– Accurately predict scatter in a deterministic framework