Concurrent structural fatigue damage
prognosis under uncertainties
Yongming Liu
School for Engineering of Matter, Transport and Energy
Arizona State University
FY13 Structural Mechanics Annual Grantee Meeting
07/24/2013, Arlington, VA
1
Outline


Background
Deterministic multi-scale fatigue damage
prognosis
- In-situ testing and multi-scale material fatigue modeling
- Concurrent structural dynamics and material damage analysis

Uncertainty management for fatigue
prognosis
- Efficient probabilistic method for real-time prognosis
- Generalized Bayesian framework for information fusion

Conclusions and future work
*This is a three year award and currently in year 3.
2
A multiscale approach for fatigue
damage analysis
Length (m)
1e-10
1e-9
1e-8
1e-6
1e-3
1e-2
1
Time
1e-15
1e-12
1e-9
1e-6
1 sec.
da/dN
Stress
(SIF)
da
dt
Time
da/dt relationship
at a smaller time
scale
years
Stress amplitude
(MPa)
Δt
days
Delta K
da/dN curve –
Paris, 1960’s
Fatigue life (N)
SN curve –
Wholer, 1860’s
loading
18
Background
16
14
Why time-based?
10
Disadvantages of the cycle-based methods





No definition for crack growth rate below one cycle
Stress ratio effects
Require cycle counting for variable amplitude loadings
Not possible for concurrent mechanical/structural analysis
and fatigue analysis
Time scale inconsistency with creep, corrosion, oxidization, etc.
8
6
4
2
0
1000
1060
1040
1020
1100
1080
Time
loading
20
18
16
14
K(MPa*m0.5)
•
K(MPa*m0.5)
12
12
10
8
6
4
2
0
•
Advantages of the time-based approach





Can capture the mechanisms at the subcyce scale
No stress ratio effects
No cycle counting requirement and direct time domain integration
Direct coupling with structural dynamics
Time scale consistent with other types of damage damage
1.02
1.03
1.04
1.05
1.06
Time
1.07
1.08
1.09
1.1
4
x 10
4
In-situ SEM testing and imaging analysis
1000
Loading
900
Unloading
800
700
N
600
500
400
300
200
100
0
0
5
10
15
20
25
30
35
40 Step
tensile sub-stage
5
Experimental results
•
•
Crack only starts to grow after a certain stress level during the loading path
crack increment (da) is directly related to the CTOD at the subcycle scale
6
5
KMAX=15.4
K
=30.7
4
KMAX=30.7
3.5
K
CTOD (um)
4
=19.2
KMAX=33.9
K
2
K
K
MAX
KMAX=30.7
3
K
MAX
Delta a (um)
5
KMAX=15.4
4.5
=33.9
MAX
=30.7
MAX
KMAX=30.7
3
K
2.5
K
K
2
=33.9
K
MAX
=19.2
MAX
1.5
=30.7
MAX
=33.9
MAX
=33.9
MAX
=33.9
MAX
1
1
0.5
0
0
0
5
10
15
20
0.5
SIF (MPa-m
Due to crack closure
25
30
35
0
5
10
)
Due to crack closure
15
20
0.5
SIF (MPa-m )
25
30
35
6
Atomistic simulation
σ yy
[010]<001>crack
• Crack growth mechanism: microvoid nucleation and linkage ahead of the main
crack tip
• Crack surface contact during the unloading path
7
Simulation results
Qualitative agreement with in-situ SEM testing
Crack growth is not uniformly distributed within one loading cycle
8
Short summary





Fatigue crack growth is a continuous phenomena at the subcycle
scale (da/dt can be defined)
Crack growth is not uniformly distributed within one cyclic loading
(no growth during the unloading path and during the initial loading
path)
The crack growth kinetics is directly related to the CTOD variation
A macro-level time-based crack growth model can be
•
•
formulated •
a = H (σ ) ⋅ H (σ − σ op ) ⋅ f (δ ) δ
The remaining question: how to efficiently track the CTOD
variation of a growing crack?
9
Analytical solution of CTOD variation
under random loading





CTOD variation under general variable amplitude loading
includes material nonlinearity (i.e., plasticity) and geometric
nonlinearity (i.e., crack surface contact)
Direct FEM simulation is very time consuming
Piecewise linearized analytical model is developed
Material non-linearity is included by a modified Dugdale
model [1] under random cyclic loading
Geometric non-linearity is included by a virtual crack
annealing model [2] assuming a certain crack length
“rewelded”
[1] Yongming Liu, Zizi Lu and Jifeng Xu, A simple analytical crack tip opening displacement approximation under
random variable loadings, International Journal of Fracture Vol. 173, Number 2 (2012), 189-201
[2] Wei Zhang, Yongming Liu, In Situ SEM testing for crack closure investigation and virtual crack annealing model
development, International Journal of Fatigue, Vol, 43, 189-196, 2012.
10
Simulation results
loading
loading
loading
15
20
18
18
16
16
14
14
10
5
K(MPa*m0.5)
K(MPa*m0.5)
K(MPa*m0.5)
12
10
8
12
10
8
6
6
4
4
2
2
0
50
150
100
Time
0
0
1000
1020
1040
1060
1080
1
1100
1.1
1.2
Single overload
-3
Christmas Tree Spectrum
crack length
x 10
-3
2.462
-3
2.04
8.46
4
crack length
x 10
2.039
2.461
2.4605
8.4
2.46
a (m)
8.42
8.38
2.038
2.037
2.4595
2.036
2.035
8.36
2.459
8.34
2.4585
2.034
2.458
2.033
2.4575
2.032
8.32
0
1.5
x 10
2.4615
8.44
a (m)
crack length
x 10
1.4
Random Spectrum
a (m)
8.48
1.3
Time
Time
1000
2000
3000
4000 5000
Time N
6000
7000
8000
9000
2.457
8900
8910
8920
8930
8940 8950
Time N
8960
8970
8980
8990
2.031
1
1.1
1.2
1.3
Time N
1.4
11
1.5
4
x 10
Validation with in situ SEM testing
Constant amplitude loading R=0.025
Constant amplitude loading R=0.1
prediction
testing data
7
CTOD (um)
6
5
4
3
2
1
0
0
10
20
SIF (MPa-m0.5)
30
40
12
Validation with CT specimen testing
20%
baseline
10%
High
ratio
Low
ratio
13
Concurrent structural dynamics and material
fatigue damage prognosis
Concurrent fatigue prognosis
Structural dynamics
••
•
m x + n x + kx = f (t )
0.4
sensor
Fatigue crack growth
•
0.3
Crack length(m)
Critical spot
0.35
2Cλ •
σa
1 − Cλσ 2
1, if x > 0
H ( x) = 
0, if x ≤ 0
•
a = H (σ ) H (σ − σ ref )
a
0.25
0.2
0.15
0.1
0.05
0
15
10
5
0
25
20
Kilocycle
Previous
rreversed
plastic
zone
p
a
Δ
a Forward plastic
r zone
during reloading
p
Research overview
IMF(1)--Mode 2
IMF(1)--Mode 1
0.01
0.08
0.008
0.06
0.006
0.04
0.02
Displacement(m)
0.02
0
-0.02
0.004
0.002
0
-0.002
-0.004
-0.04
-0.006
-0.06
-0.08
4
x 10
-3
-0.008
0
0.5
1
Time(sec)
IMF(1)--Mode 4
1.5
-0.01
2
0
0.5
1
1.5
-3
Time(sec)
x 10
5
0
2
IMF(1)--Mode 3
4
3
3
-0.02
-0.06
-0.08
0
imf #2
2
0
-2
imf #3
2
0
-2
imf #4
0.01
0
-0.01
5
0
-5
0
-3
x 10
0
-3
x 10
0
-4
x 10
0
0.5
1000
1000
1000
1000
1
Time(sec)
2000
2000
2000
2000
3000
3000
3000
3000
1.5
4000
4000
4000
4000
2
φ11 φ12 
φ φ 
 21 22 
φ11 δ11
=
φ21 δ 21
1
0
-1
-3
-4
-3
-4
-5
0
0.5
1
Time(sec)
1.5
2
5000
5000
5000
5000
3
Mode
matrix
x 10
0
0.5
1
Time(sec)
1.5
-3
Theoretical displacement
Extrapolated displacement
2
unknown
place
1
0
-1
-2
-3
Mode extraction
2
1
0
-1
-2
-2
Displacement(m)
-0.1
Displacement(m)
2
-0.04
Displacement(m)
Displacement(m)
0.06
Displacement(m)
sensor
data
0.1
0.08
imf #1
0.04
Extrapolation process for the
unknown spots
1
1.1
1.2
1.3
Time(sec)
1.4
1.5
1.6
Dynamic response14
reconstruction
2
Strain-based dynamic response
reconstruction
Strain gauge measurements
Location of interest
Element
e
u
Measurement ε ( e ) (t )
1
Decomposition
(EMD)
ε ( e ) (t ) ≈
Modal responses
ηi( e ) (t ), i = 1...m
Reconstructed ε (u ) (t )
∑η
i =1...m
(e)
i
(t )
ε ( u ) (t ) ≈
∑η
i =1...m
2
Transformation
 B ( e )Φ i( e ) 
(u )
(e)
ηi (t ) = ηi (t ) (u ) (u ) 
 B Φi 
(u )
i
(t )
3
Superposition
Modal responses
−1
ηi(u ) (t ), i = 1...m
15
Demonstration example - description
Random pressure
-3
1
x 10
Strain
0.5
0
1
2
3
4
5
6
8
7
-0.5
0.05m
2
-3
4
6
Time (second)
8
10
strain measurement
(a)
Loc. 1
Strain
measurements
x 10
0.2
0.4
0.6
Time (second)
0.8
1
strain measurement
(b)
0
-1
5
-4
Fourier spectra
Identified frequencies
1
Modal 2
x 10
x 10
-4
x 10
Loc. 3
-4
x 10
0
-5
-4
0
-5
0.5
5
x 10
Modal 3
-1
0
Modal 1
1
-0.5
Amplitude
-5
-3
0
Loc. 2
hotspots
2
Modal 4
Strain
0.5
1.5
10
5m
-1
0
1
9
-4
x 10
1
0
-1
4
4.5
5
Time (second)
5.5
6
Four modal responses of the strain measurements
16
data obtained using EMD
Fourier spectra of(c)the measurement data
0
0
20
40
60
Frequency (Hz)
80
100
Demonstration example – verification of
strain reconstruction
Random pressure
-3
1
Theoretical
Reconstructed
x 10
Strain
0.5
1
2
3
4
5
6
7
8
5m
-0.5
Loc. 1
-1
5
5.2
5.4
5.6
Time (s)
5.8
6
(a)
Strain
measurements
Loc. 2
hotspots
-3
-3
1
Theoretical
Reconstructed
x 10
Loc. 3
( )
Results for Loc. 1
Theoretical
Reconstructed
x 10
0.5
Strain
0.5
Strain
10
0.05m
0
1
9
0
0
-0.5
-0.5
-1
5
-1
5
5.2
5.4
5.6
Time (s)
(b)
Results for Loc. 2
5.8
6
5.2
5.4
5.6
Time (s)
(c)
Results for Loc. 3
5.8
6
17
Demonstration example – verification of
crack growth prediction
0%noise
9
5%noise
10%noise
Crack length (mm)
2%noise
8
7
Theoretical
1% RMS noise
2% RMS noise
5% RMS noise
10% RMS noise
6
5
0
50000
100000
150000
Time (s)
18
Efficient probabilistic fatigue crack growth
analysis

Tremendous uncertainties for fatigue crack growth
analysis in service conditions
- future unknown loading uncertainty (e.g., amplitude, sequence)
- material property uncertainty (e.g., initial crack length, FCG
coefficients)


Classical Monte Carlo (MC) simulation is not
appropriate for concurrent analysis due to its
computational cost
Efficient probabilistic analysis framework
- equivalent random loading transformation
- Inverse First Order Reliability Method (iFORM)
19
Equivalent random loading transformation
– non-coupling case

Key idea: matching the final life prediction of a realistic random
loading to an equivalent constant loading
Δσ2
Stress
Δσ1
150
0.08
100
0.06
f(x,y)
100
Loading (MPa)
200
50
0
0.04
0.02
-50
0
-3
-2
-100
-1
0
-150
0
10
20
30
40
50
Faituge life (cycles)
0
0
1
Loading
2
Same Life
150
0.021
Stress
1
n
= (∑
1
0
2
N i g ( Ri )
∆σ )
N g (0)
1
g ( Ri )
pi ( Ri , ∆σ i )
∆σ im ) m
g (0)
Crack Length (m)
n
∆σ eq = (∑
1
m m
i
Random loading process
Equivalent stress level
0.019
0.018
0.017
0.016
0.015
0.014
20
0.013
0.012
0
1000
2000
3000
Fatigue life (cycles )
4000
5000
10
Stress range (MPa)
0.02
Δσeq
1
Loading
300
200
0.022
100
0
1
Stress ratio
250
Equivalent random loading transformation
– coupling case

Key idea: using the time-based subcycle FCG model to calculate
a correction factor distribution for the coupling effect
∆σ eq = η∆σ eq
*
Fitting curve results, z
1
0.9
0.8
0.7
0.6
0.5
1
1.7
Forward plastic zone
200
0
Occurrence probability, y
Coupling zone
a
1.3
1.2
Overload ratio, x
Gaussian Process
modeling
100
a0L
50
0
1
rp
-50
(rp)0L
rpv
0.9
empirical CDF
0.8
Fitting results
0.7
10
20
30
40
50
Fatigue life (cycles)
Random loading
0.6
Time-based FCG
model
F(x)
Loading (MPa)
1.5
1.4
150
-100
0
1.6
0.5
Reversed plastic zone
0.5
0.4
0.3
0.2
0.1
0
0.6
Probability Distribution of η
0.7
0.8
0.9
1
1.1
1.2
21
Efficient probabilistic fatigue life prediction
algorithm - iFORM
Key idea: calculate the remaining useful life (RUL) at a specified
reliability level by solving the inverse reliability problem

Input/quantified
uncertainties
Output/propagated
uncertainties
1.4
1.2
1.4
1
MPP
0.8
1.2
1
0.6
0.8
0.4
0.6
0.2
0.4
0
-2.5
-2
-1.5
-1
Stress ratio
-0.5
0
0.5
0.2
0
-2.5
-2
-1.5
-1
Stress ratio
-0.5
0
0.5
1.4
1.2
Mechanism
model
1
0.8
0.6
0.4
0.2
0
-2.5
-2
-1.5
-1
Stress ratio
-0.5
0
Probabilistic RUL
Estimation
0.5
22
Demonstration and validation of
probabilistic prognosis

7 CT specimen testing under stationary random loading
spectrum for FCG testing
16
Pmax,OL
CT-VI-2
n1
n2
n3
Crack length (mm)
Pmax
CT-VI-1
14
n1
12
CT-VI-3
CT-VI-4
10
CT-VI-5
CT-VI-6
8
CT-VI-7
Median prediction
6
Pmin,
95% confidence bounds
Time
Specimen Loading sequence*
CT-VI-1 n1=23, n2=10, n3=14
CT-VI-2
n1=42, n2=3, n3=2
CT-VI-3 n1=16, n2=16, n3=15
CT-VI-4 n1=30, n2=10, n3=7
CT-VI-5 n1=27, n2=20, n3=0
n1=47, n2=0, n3=0
CT-VI-6
CT-VI-7 n1=12, n2=20, n3=15
Equivalent Stress
model + iFORM
4
Inverse FORM
2
0
10
20
30
40
Fatigue life (cycle)
Approach
Point-by-point FCG simulation
(+MC simulation )
Equivalent stress level
(+MC simulation)
Equivalent stress level
(+Inverse FORM)
50
60
70
Thousands
Computational time
~10 hours
2309 seconds
5 seconds
23
Conclusion and Future work
 Integrated experimental and simulation framework for
multiscale fatigue analysis
 New, systematic, and alternative time-based subscycle
FCG formulation
 Concurrent structural dynamics and material fatigue
damage analysis
 Efficient probabilistic prognosis under random loadings
 Meso-scale modeling for fatigue crack growth simulation
 Generalized information fusion framework for risk
assessment and decision making under uncertainties
Publications – journal articles








Zhang, W. and Y. Liu (2012). "In situ SEM testing for crack closure investigation and virtual
crack annealing model development." International Journal of Fatigue 43(0): 188-196.
He, J., X. Guan, et al. (2012). "Structural response reconstruction based on empirical mode
decomposition in time domain." Mechanical Systems and Signal Processing 28(0): 348-366.
Lu, Z. and Y. Liu (2011). "Experimental investigation of random loading sequence effect on
fatigue crack growth." Materials & Design 32(10): 4773-4785.
Jian Yang, Wei Zhang, Yongming Liu, Existence and Insufficiency of the Crack Closure for
Fatigue Crack Growth Analysis, International Journal of Fatigue, 2013. (conditionally
accepted)
He, J., Guan, et al. (2013). “Time domain strain and stress reconstruction for concurrent
fatigue damage prognostics” Mechanical Systems and Signal Processing (under review)
Xiang, Y., Liu, Y. An Equivalent Stress Transformation for Efficient Probabilistic Fatigue
Crack Growth Analysis under Variable amplitude Loadings, ASCE Journal of Aerospace
Engineering, 2013. (under review)
Zhang, W. and Y. Liu . “Time-based subcycle fatigue crack growth modeling. Part I:
analytical approximation for crack tip displacement”, International Journal of Fatigue.
(under preparation)
Zhang, W. and Y. Liu . “Time-based subcycle fatigue crack growth modeling. Part II: crack
growth simulation and validation”, International Journal of Fatigue. (under preparation)
25
Publications – conference
proceedings and presentations









Enqiang Lin, Hailong Chen, Yongming Liu, “atomistic simulations of fatigue crack growth in single crystal
aluminum”, ASME 2013 International Mechanical Engineering Congress & Exposition, San Diego, CA.
Jian Yang, Wei Zhang, Yongming Liu, “Subcycle fatigue crack growth mechanism investigation for
allumnium alloys and steels”, International Conference of Fracture, 2013, Beijing, China.
Jingjing He; Xuefei Guan; Yongming Liu, "Concurrent structural and material fatigue damage prognosis
integrating sensor data", AIAA SDM conference, 2013, Boston, MA.
Wei Zhang; Yongming Liu, "A time-based formulation for real-time fatigue damage prognosis under
variable amplitude loadings", AIAA SDM conference, 2013, Boston, MA.
Jian Yang; Wei Zhang; Yongming Liu, "Subcycle Fatigue Crack Growth Mechanism Investigation for
Aluminum Alloys and Steels", AIAA SDM conference, 2013, Boston, MA.
Wei Zhang, Yongming Liu, “Subcycle fatigue damage mechanism investigation using In-situ SEM testing”,
ASME 2012 International Mechanical Engineering Congress & Exposition, Houston, TX.
H. Li, Y. Xiang, Y. Liu, “Probabilistic fatigue life prediction using Subset Simulation”, 53rd Structures,
Structural Dynamics, and Materials and Co-located Conferences, Honolulu, Hawaii , April, 2012.
W. Zhang, Y. Liu, “In-situ fatigue testing on the existence and insufficiency of the crack closure”,
International Conference on Fatigue Damage of Structural Materials IX, Hyannis, MA, USA, September,
2012.
Jian yang, wei zhang, yongming liu, “A Multi-Resolution Experimental Methodology for Fatigue Mechanism
Verification of Physics-Based Prognostics”, Annual Conference of the Prognostics and Health Management
Society 2012. ISBN: 978-1-936263-05-9
26
Acknowledgements


The research reported was supported by funds from
Air Force Office of Scientific Research (AFOSR) Young Investigator Program (Contract No. FA955011-1-0025, Project Manager: Dr. David Stargel). The
support is gratefully acknowledged.
Discussion and encouragement from Eric Tuegel, Reji
John, and Ravi Penmetsa at AFRL.
27
Thanks!
Questions?
28