University of Illinois at Urbana-Champaign
Department of Mechanical Science and Engineering
Recent Advances in Modeling Fatigue in Metals
H. Sehitoglu, M. Sangid,T. Ezaz, H.J.Maier
University of Illinois, USA, University of Paderborn, Germany
Symposium in honor of C. Tome
TMS, San Diego, February 28, 2011
Work Supported by Rolls Royce, National Science Foundation, DMR, Metals
1
www.mechse.uiuc.edu
Outline
Modeling 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 Initiation
Experimentally 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 Simulations
Use 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 
N FCC
E Perfect
M
Area
GB
ECSL

Atomic Simulations of Prevalent Tilt & Twist GBs
Grain Boundary Energy (mJ/m2)
1600
<110> Tilt
<111> Twist
<001> Tilt
1400
Σ5
1200
1000
Σ19
Σ5
Σ9
<001>
Σ17
θ(
800
600
Σ13
Σ7
Σ11
Σ21
400
200
Perfect
FCC
0
0
60
Σ3
90
<110>
Σ7
Σ21
Σ21
Σ3
30
Σ13
Σ13 Σ7
Perfect
FCC
Σ3
120
Rotation Angle (degrees)
150
θ
180
(
<111>
Mechanical Behavior of GBs: Slip Transmission
A Σ3 (Twin) GB
a/2 [011] t
(100)t
a/6
[211](111)
Y [111]
X [211]
Z [011]
a
a
a
211  011T  100
6
2
3
Incident
Lomer
Stair Rod
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
GB Energy Barriers to Slip
•
Monitor the energy of the atoms within a
control box at the GB
n
E
•
E
i
load
 Estatic
i
volum e
Obtain the energy barrier for slip to penetrate
the GB
Validated values of slip in a perfect FCC lattice
by comparing with GSFE curve
•
400
Static GSFE Curve
Control Box Method
DFT Data from Siegel, 2005
Experimental Data
Stacking Fault Energy (mJ/m2)
350
300
250
200
150
100
Error ~ 6%
50
0
0
0.5
1
1.5
Reaction Coordinate: Uz/<112>/6
2
Measured Energy Barrier for Slip Transmission for various GBs
Validation of MD Results of Slip Transmission
Criterion for Slip Transmission
•
Lee, Robertson, and Birnbaum, 1989
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 ┴):
Energy Barriers for Slip Transmission
through GB
Energy Barrier for Dislocation - GB
Interaction (mJ/m3)
2.5E+12
Σ3
2.0E+12
E
Transmissi on
Barrier
1.5E+12
Σ7
Σ11
1.0E+12
Σ13
Σ21
Σ17
Σ5
5.0E+11
Σ9
Perfect
FCC
Σ19
0.0E+00
0
200
400
600
800

 2.8 10  E
13
1000
Static GB Energy (mJ/m2)
1200
1400

0.6
GB
Static
Twin boundary has inherently high energy
barriers for slip transmission
Coherent twin boundary
• Stable defect structure
• Low static GB energy
• High energy barrier for
slip transmission
Note:
• Applied loading is normal to GB
• No applied shear stress on GB
• General twin-slip interaction
energies please refer to:
Ezaz T, Sangid MD, Sehitoglu H,
“Energy barriers associated with sliptwin interactions,” Philosophical
Magazine, (2011).
DOI: 10.1080/14786435.2010.541166
Slip is initially impeded by
the twin resulting in a
dislocation pile-up.
As the applied load
increases, slip transmits past
the twin boundary.
Measured Energy Barrier for Slip Nucleation from various GBs
Energy Barriers for Slip Nucleation from GB
Energy to Nucleate a Dislocation (mJ/m3)
2.0E+12
Σ7
1.8E+12
Σ21
Σ13
1.6E+12
Please 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
Σ17
1.4E+12
1.2E+12
1.0E+12
8.0E+11
Σ9
Σ5
6.0E+11
Σ19
4.0E+11
2.0E+11
0.0E+00
0
200
400
600
800
1000
Static GB Energy (mJ/m2)
1200
1400
Fatigue Crack Initiation
Experimentally 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
Applied Work
Extrusion Formation at GBs
Dislocation Nucleation at GBs
Shearing of γ’ Precipitates
Exp functions
σ ≡ Applied stress
h ≡ Width of PSB
d ≡ Dislocation spacing
ρ ≡ PSB dislocation density
N ≡ Number of cycles
Microstructure Inputs
where:

Output
Dislocation Pile-ups
disl
slipGB
Enuc
(m,,h,L,L',N)  Eextrusion
(m,,h,L,L',N)  EAPB (L,dist ,N)  E SF (L,dist ,N)
Continuum

Work Hardening in Bands
MD
disl
Energy Eapp (,m,L,N)  Ehard (,L,N)  E pileup
(h,d,L,N) 
Monitor a PSB and when it reaches a
stable point, the material fails!
m ≡ Schmid factor
L ≡ Grain size
Σ ≡ Characteristic of GB
γ’ ≡ Distribution of precipitate
L’ ≡ Grain size of neighboring grain
Each term is expressed in terms
of a slip increment, ∂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).
Shearing of γ Matrix
Energy associated with overcoming stress field within
the PSB for motion of glissile dislocations
r layers
E   bLn X
where the total stress is given by:
  
dis
Total stressPile-up of
dislocations
 
A
Applied
stress
h
where:
b = Burgers vector
L = grain size
nlayers = number of dislocation
layers in the PSB related to width
of PSB, normalized by annihilation
distance
Lattice
resistance
∂X = increment of slip
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 Energy
Atomistic Based Formulation:
• Dislocations nucleate from the GB and localize
in slip bands
disl
nuc
E
  Xi  EMD
 nucGB
r 2
  o bhL
i
– The number of dislocations that are emitted
from the GB and aggregate within the PSB is
given by:
  o hL
• PSBs form extrusions at grain boundaries in
polycrystalline material
slipGB

extrusion
E
  Xi  EMD
  slipGB
r
n bh
pen
dis
i
• Leverage energy barriers for slip transmission
and nucleation at a GB from atomic
simulations (previously shown)
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
Stacking Fault Energy +
Anti-Phase Boundary Energy
Atomistic Based Formulation:


layers
EAPB  E SF  f   APB dL  1 f   SF dL neff
X
d
d
o
o
where f is the area fraction of γ’, fU720 ~ 0.20
Shearing of γ’
precipitates
O
M
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).
R
M’
O’
PSB Energy Balance
Sangid MD, Ezaz T, Sehitoglu H,
Robertson IM, “Energy of slip
transmission and nucleation at grain
boundaries,” Acta Materialia 59 283296 (2011).
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).
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:
pen
layers
h, d ,  A , , ndis
, nlayers , neff
• Criterion for crack initiation
– Energy of PSB reaches a stable value
– Minimize energy to check for stability of PSB:
E
0
X i
where Xi represents the position of the glissile (mobile)
dislocations in our energy expression
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
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
TEM pictures from: Zhang & Wang, Acta Mat 51 (2003).
Clustering of grains
Failure occurs due to PSB-GB interaction
or
Grain cluster (multiple grains connected
by LAGBs)
Single large grain
Normalized Applied Strain Range, %
1.2
Fatigue Scatter Results
Model - Simulated Specimens
Model - Average
U720 Experimental Data
U720 Data - Average
1.1
1
1000 simulated specimens vs. 84
experimental results
0.9
Each simulated specimen takes <30
seconds to construct its
microstructure and predict fatigue
life for a series of strain ranges
0.8
0.7
0.6
100
1000
10000
100000
1000000
Cycles to Initiation
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