1. MultiStage Fatigue (MSF) Modeling
Dr. Mark F. Horstemeyer (Mississippi State University)
Outline
Introduction/motivation
Micromechanics: Computations and experiments
MultiStage Fatigue (MSF) model
Summary
Main Reference
McDowell, D.L., Gall, K., Horstemeyer, M.F., and Fan, J., “Microstructure-Based Fatigue Modeling of Cast A356-T6 Alloy,” Engineering Fracture Mechanics, Vol. 70, pp.49-80, 2003.

3. initial microstructure-
ISV-MSF Model Implementation/Use
mesh
initial microstructure-
inclusion content
finite
element
Code
(ABAQUS)
MSF
Model
life
ISV
model
Damage/failure
boundary conditions
loads
temperature
strain rate
history
design
Note: models can
be implemented
in other FE codes

4. MSU MSF Model History
First started on a cast A356 al alloy for automotive application ( )
Extended to aerospace aluminum alloys (7075, 7050 al) ( )
Extended to automotive cast Mg alloys (2002-present)
Recently used for several steel alloys (2005-present)
Just started polymers
McDowell, D.L., Gall, K., Horstemeyer, M.F., and Fan, J., “Microstructure-Based Fatigue Modeling of Cast A356-T6 Alloy,” Engineering Fracture Mechanics, Vol. 70, pp.49-80, 2003.

5. MSU MultiStage Fatigue Modeling
Based upon three thresholds
Incubation
Microstructurally Small Crack Growth
Long Crack Growth
Based on microstructure sensitivity
Multiscale modeling was used to first develop the equations in the absence of experiments; experiments later validated the equations

6. MultiStage Fatigue Microstructure-Sensitive Model
Ntotal=Ninc+NMSC+NPSC+NLC
Ntotal = total number of cycles to failure
Ninc = number of cycles to incubate a fatigue crack
NMSC = Microstructurally Small Crack growth (ai < a < kDCS)
NPSC = Physically Small Crack growth (~1-2DCS < a < ~10DCS)
NLC = Long Crack growth (a > ~10DCS)
Inclusion Severity
1. Large oxides greater than 200 microns
2. Large pores near free surface (length scale ~ 100 microns)
3. Large pores (length scale ~ Microns)
4. High volume fraction of microporosity; no large pores/oxides
(length scale < 50 microns)
5. Distributed microporosity and silicon; no significant pores/oxides

7. Ilustration of Different Stages

8. Different Defects Induce Different Crack Growth Rates

11. Strain-Life Data

12. Fatigue Micromechanisms
LCF and HCF Regimes.
Mechanisms
LCF - Extensive
Plasticity
HCF – Microstructure Scale Plasticity
Crack incubation
largest grains or inclusions establish initial crack length in propagation analysis
Initiation-dominated: largest grains or inclusions control number of cycles to form a crack or to propagate past arrest limits
MSC growth
Cracks grow in elastic-plastic field with less microstructure influence
First few microstructural barriers control fatigue limit and scatter of lifetime
PSC
and LC growth
Elastic-plastic growth persists well into crack growth history; coalescence of multisite cracks can occur
Transition to LEFM-dominated homogeneous crack growth; single dominant crack is common

13. Fatigue Stages: Incubation (b)
Fatigue damage of AA 7075-T651 was found mostly initiated at fractured particles
NINC: The number of cycles required to nucleate a crack at a constituent particle and then to grow the crack a short distance from the particle; in this state, the fatigue damage evolution is under the influence of micronotch root plasticity.
NINC uses modified Coffin-Manson law
: micro-notch root max plastic shear strain
a : Remote Strain; l : plastic zone size
D : particle diameter; R : min/max
ainc = 0.5 Dp + 1/16 Dp, the crack size is 2ainc
Experiments/Simulation for Incubation Life
Measurement/evaluation of notch root plastic strain amplitude :
2-D micromechanics simulation of fractured particles for local plasticity as a function of remote loading (MSU)
Conducting interrupted HCF tests in-situ SEM on polished rectangular specimens with laser cut micronotch of particles (MSU)
Measure at micron scale the local plastic strain (amplitude and plastic zone size) using Micro-X-Ray diffraction to evaluate the micronotch plasticity to understand/validate the incubation model (ORNL)

14. Incubation (Ninc) Measurement/evaluation of Incubation Life NInc:
: micro-notch root max plastic shear strain
a : Remote Strain; l : plastic zone size
D : particle diameter; R : min/max
ainc = 0.5 Dp + 1/16 Dp, the crack size is 2ainc
Measurement/evaluation of Incubation Life NInc:
In-situ SEM fatigue test using dogbone shape rectangular specimens with micronotches to observe the crack incubation and growth with R = -1, 0.1, This provides accurate incubation life prediction and crack size and crack growth rate measurement to submicron scale. (MSU)
Single Edge Notch Tension tests (SENT) with R = -1, 0.1, 0.5 observation on small crack initiation and propagation using 1) optical tools (~50 mm), 2) plastic repliset (~10 mm). This provides incubation life estimation and crack size and crack growth rate measurement to micron scale. (MSU, for FASTRAN as well)
Interrupted strain-life fatigue experiments with R=-1,0.1 on Kt  3 specimens (previous done at Alcoa) that estimate incubation life as a function of stress states

15. Fatigue Incubation Indicators
Exhaustion of irreversible strain (slip band decohesion):
Suresh, 1990
cf. Dunne et al.
Irreversibility factor or

16. Fatigue Incubation Indicators
Modified Coffin-Manson laws for crack formation (incubation), assuming cyclically stable conditions:
cf. Mura et al. (1991)
Fatemi-Socie Parameter (1988)  decohesion plus crack behavior
(McDowell & Berard, FFEMS, 1992)
(cf. Dang-Van (1993), Papadopoulos (1995), others for similar
multiaxial parameters applied at grain scale)

17. Fatigue Incubation Indicators
Zener mechanism or
Stress normal to boundary

18. Incubation life NINC = maximum plastic shear strain range
at particle/matrix interface averaged in a
process zone volume
Refs
1 Coffin-Manson
2. Venkataraman et al., 1991
3. Dowling, 1979
4. Ting and Lawrence, 1993
5. McDowell et al., 2003
RHS-constants correlated from
uniaxial fatigue exps
LHS-constants determined from
micromechanical FE simulations

19. Solving for Right Hand Side of Incubation Eqtn: Partition of HCF/LCF based upon local Plasticity
HCF strain coefficient at
micronotch
LCF strain coefficient at
micronotch
Threshold between constrained and unconstrained
microplasticity determined from micromechanical
FE sims
Refs
McDowell et al., 2003
Gall et al., 2000
Gall et al., 2001

20. Solving for Right Hand Side of Incubation Eqtn: HCF Mean Stress Effect
Local microstructure-based fatigue ductility coefficient
Cn ~ material constant 0.2 ~ 0.6 (Cn=0.48)
Cm ~ material constant ~ 1.0 (Cm=0.3)
C-M Fatigue ductility exponent
~ material constant ~ -0.9 (a = -0.7)

21. Solving for Left Hand Side of Incubation Eqtns: transfer functions needed
Micromechanical simulations relate global applied strain range to maximum plastic shear strain range at particle/matrix interfaces
Refs
McDowell et al., 2003
Gall et al., 2000
Gall et al., 2001

22. Solving for Left Hand Side of
Incubation Eqtns: Micro FE Sims

23. Solving for Left Hand Side of Incubation Eqtns: Micro FE Sims
eper=strain percolation limit for microplasticity at inclusion ( , eper= )
Determined by cyclic yield strength (=0.8Sy/E(1-R))
Determined by micromechanical FE sims
Determined by ORNL micro X-ray diffraction method
eth=strain threshold for microplasticity inclusion ( , eth=0.0021)
Determined by Su of material (=.29Su/E/(1-R))
Determined by fatigue strength (=Sf/E)

24. Solving for Left Hand Side of Incubation Eqtns: Micro FE Sims
hlim=l/D at the strain percolation limit ( , hlim=0.3) determined by micromechanical FE sims
r=l/D exponent ( , r=0.4) determined by micromechanical FE sims
q=nonlocal microplastic shear strain range exponent ( , q=2.27) determined by micromechanical FE sims
Y1=nonlocal microplastic shear strain range coefficient ( ,Y1=116) determined by micromechanical FE sims
Y2=nonlocal microplastic shear strain mean stress coefficient ( , Y2=0) determined by micromechanical FE sims
x=strain intensification multiplier (1-9, x=1.6) determined by micromechanical FE sims

25. size of incubated crack
Refs
Smith and Miller, 1977
McDowell et al., 2003

26. When does the transition occur between stages?
Incubation (Current method has more influence on HCF than LCF)
MSC (current method assumes long crack starts at 250 microns)

27. Fatigue Stages: MSC/PSC
NMSC/PSC : the number of cycles required for a microstructurally small crack and physically small crack propagating to a long crack; in this state, the crack growth are influenced by microstructural noncontinuous features, such as particle, particle distribution, grain size and orientation, and textures.
Fatigue Model
Multiaxial term

28. MSC Regime’s Different Plasticity Character

29. MSC Regime (Grain effects)
Multiaxial term
Crystal plasticity fatigue simulation on crack propagation validate grain orientation effects (MSU or Cornell)

30. MSC Regime (CIII)
Crystal plasticity fatigue simulation on crack propagation overload or load sequence effects
Periodic overload experiments for Kt=1 specimens
Sequence experiments for Kt=1 specimens

31. MSC Regime (CI and CII)
In-situ SEM fatigue test using dogbone shape rectangular specimens with micronotch to observe the crack initiation and growth with R = -1, 0.1, 0.5
Single Edge Notch Tension tests (SENT) with R = -1, 0.1, 0.5 observation on small crack propagation using 1) optical tools (~50 mm), 2) plastic repliset (~10 mm). This provides MSC life estimation and crack size and crack growth rate measurement to micron scale. (MSU, for FASTRAN as well) (LaVision system)
Sequence experiments for Kt=1 specimens
Periodic overload experiments for Kt=1 specimens for just CI

32. MSC Regime (q1 and q2) Multiaxial term
Multi-axial tests to determine 1 and 2.

33. MSC CTD Drops Indicate Resistance from Particles

34. MSC Showing Tortuousity via FEA
Fan, McDowell, Horstemeyer, and
Gall, K.A., Eng Fract Mech, 68,
No. 15, pp , 2001.
Resistance of particles and pores to
small cracks is illustrated

35. MSC

36. Microstructurally Small Crack, NMSC
crack growth rate is a function of crack tip displacement range
G ~ constant for given microstructure with 0.30 < G < 0.50
G=0.32 for 7075 al alloy
G is being evaluated from Crystal Plasticity and atomistic sims
DCTDTH ~ Burgers vector b
Refs
Laird et. al., 1965
McClintock, 1965
3. McDowell et al., 2003

37. DCTD calculation HCF LCF
Refs
Dugdale
Couper et al., 1990
Shiozawa et al., 1997
McDowell et al., 2003
HCF
LCF
GS = grain size (19-74 microns, GS=40), determined by CMU
Su = ultimate strength (635 MPa) determined by NGC exps
n = MSC HCF exponent ( , n=4.24) determined by small crack exps
a = crack length
CI=MSC LCF Coefficient (1e4-6e4 microns, CI=1.6e4) determined by in-situ SEM (now it is determined by strain-life exps)
CII= MSC HCF Coefficient ( , CII=1.82) determined by in-situ SEM (now
It is determined by strain-life exps)
w=Hall-Petch fatigue exponent (0-1, w=0)

38. U considers crack closure
simple approximation
Refs
McDowell et al., 2003
Fan et al., 2001
R < 0 So = 0 U = 1/(1-R)
R > 0 So = Smin U = 1
So is determined by small crack mean stress
experiments, in-situ SEM, and
micromechanical crack growth FE sims

39. Multiaxial stress effects
deviatoric von Mises stress
Refs
McDowell et al., 2003
Hayhurst et al., 1985
maximum principal stress
0 < q < 1
q = 0 fatigue controlled by
q = 1 fatigue controlled by
q determined by torsion-tension/compression fatigue exps

40. Long crack growth NLC
FASTRAN used for long crack growth

41. Transition from small crack growth to long crack growth

42. initial microstructure-
Use of Fatigue Model
mesh
initial microstructure-
inclusion content
finite
element
Code
(ABAQUS)
Number of
cycles
to failure
Fatigue
model
Note: coupon tests from a component
are typically uniaxial, but the stress state
of a region in the component is typically
multiaxial
boundary conditions
loads
temperature
strain rate
history

43. Notch Root Radii Effects on Incubation and MSC

44. MSC:
Debonding dominant
because driving force
is relatively small
LC:
Cracking of second
phase particles dominant
because driving force
is relatively strong

45. Strain-life for A356 Al alloy with a focus on local defects

46. Fracture surface of 0.2% strain amplitude sample Fatigue crack
Nucleation site
Al oxide
cavity where the growth of
microstructurally small cracks occurred
Fracture surface
of 0.2% strain
amplitude
sample
specimen surface

47. FCG =fatigue crack growth
Same fracture surface of 0.2% strain amplitude sample as before showing progressive damage
alpha intermetallics
FCG =fatigue crack growth

49. SEM pictures at (a) 15x and (b) 200x of specimen tested under uniaxial fatigue at a
strain amplitude of with an R-ratio of –1. This specimen ran for 2.05x106 cycles
illustrating the degrading effect of the 150 micron size casting pore.

50. SEM pictures at (a) 15x and (b) 200x of specimen tested under uniaxial fatigue at a
strain amplitude of with an R-ratio of –1. This specimen ran for 51,000 cycles
illustrating the degrading effect of the 100 micron size casting pore at the specimen edge.

51. Number of cycles versus maximum pore size (micron) measured for specimens
tested at a strain amplitude of

52. Number of cycles versus nearest neighbor distance (micron) measured for specimens
tested at a strain amplitude of .

53. Number of cycles versus number of pores measured for specimens tested at a
strain amplitude of

54. Number of cycles versus porosity (void volume fraction) measured for specimens tested at a strain amplitude of

55. Number of cycles versus (pore size
Number of cycles versus (pore size*pore size)/(nearest neighbor distance*dendrite
cell size) measured for specimens tested at a strain amplitude of

56. Current State: Multistage Fatigue Model
Incubation
Initial crack size
MSC/PSC Growth
Note: not used
for PM alloys
HCF loading dominated
LCF loading dominated
Multiaxial term
Mean stress term
Porosity term
LC Growth
LC growth model will be FASTRAN. This model is temporary.

57. Used to determine functional
form of incubation equation
particularly the 0.3 limit

58. Micromechanics simulations showing variation
of driving force because of pore/particle resistances

59. Strain-Life as a function microstructure
Long Crack Regime

60. Number of Cycles as a function of inclusion size

62. Strain-Life Model Correlation with MSF Model

63. Finite Element Analysis of Performance and Fatigue
Total Fatigue Life NTOTAL = NINC + NSC + NLC for Higher Bound (Low Homogenous Porosity 9.5%)
NT > 10,142,944 cycles
NT > 10,106,046 cycles
20,000 lbs
21,000 lbs
NT > 10,079,826 cycles
NT > 10,060,891 cycles
22,000 lbs
23,000 lbs

64. Powder Metal Finite Element Analysis of Performance and Fatigue
Higher Bound
(Low Homogeneous Porosity 9.5%)
Fatigue Life
Shaft Loading (lbs)
20,000
21,000
22,000
23,000
NINC
> 10,000,000
NSC 2 1
NLC
142,942
106,044
79,826
60,890
NTOTAL
> 10,142,944
> 10,106,046
> 10,079,828
> 10,060,891
Failure
PASS
Lower Bound
(High Homogeneous Porosity 19.0%)
Fatigue Life
Shaft Loading (lbs)
20,000
21,000
22,000
23,000
NINC
2,399,554
1,658,665
1,176,098
778,410
NSC 1
NLC
173,956
129,105
97,223
74,187
NTOTAL
2,573,511
1,787,771
1,273,322
852,598
Failure
FAIL
Interpolation I
(Heterogeneous Porosity)
Fatigue Life
Shaft Loading (lbs)
20,000
21,000
22,000
23,000
NINC
> 10,000,000
9,902,806
8,341,563
7,109,965
NSC 1
NLC
163,212
121,115
91,194
69,579
NTOTAL
> 10,163,213
10,023,922
8,432,758
7,179,545
Failure
PASS
CRACK
FAIL
Interpolation II
(Heterogeneous Porosity)
Fatigue Life
Shaft Loading (lbs)
20,000
21,000
22,000
23,000
NINC
> 10,000,000
8,553,476
7,485,101
NSC 1
NLC
163,212
121,115
91,194
69,579
NTOTAL
> 10,163,213
> 10,121,116
8,644,671
7,554,681
Failure
PASS
FAIL

65. Relationship of Manufacturing Process, Defect, and Fatigue Mechanisms
Rolling/Extrusion/Forging/Stamping
Particles
15% INC
70% MSC
15% LC
Manufacturing process
Defect type
Dominant damage mechanism
under cyclic loads
Casting
Particles Porosity
25% INC % INC
65% MSC % MSC
10% LC % LC
N=Number of Cycles
NINC=Incubation
NMSC=Microstructurally Small Crack
NLC=Long Crack
N=NINC+NMSC+NLC
Powder metal compaction/sintering
Porosity
99% INC
0% MSC
1% LC
Defect size (m)
10-7
10-6
10-5
10-4
10-3
Fatigue
Failure
Defect volume
fraction
10-5
10-4
10-3
10-2
10-1
10-0