1

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

2

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.

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. days years

A multiscale approach for fatigue damage analysis

Fatigue life (N)

Str

ess

ampl

itude

(MP

a)SN curve – Wholer, 1860’s

Delta K

da/d

N

da/dN curve – Paris, 1960’s

Time

Stre

ss(S

IF)

dt

da

Δt

da/dt relationship at a smaller time scale

4

• 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.

• 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

1000 1020 1040 1060 1080 11000

2

4

6

8

10

12

14

16

18loading

Time

K(M

Pa*

m0.

5 )

1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1

x 104

0

2

4

6

8

10

12

14

16

18

20loading

Time

K(M

Pa*

m0.

5 )

Background

Why time-based?

In-situ SEM testing and imaging analysis

5

tensile sub-stage

0100

200300

400500

600700

800900

1000

0 5 10 15 20 25 30 35 40 Step

N

Loading

Unloading

6

0 5 10 15 20 25 30 350

1

2

3

4

5

6

SIF (MPa-m0.5)

CTO

D (u

m)

K

MAX=15.4

KMAX

=19.2

KMAX

=30.7

KMAX

=30.7

KMAX

=30.7

KMAX

=33.9

KMAX

=33.9

KMAX

=33.9

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

Due to crack closure

0 5 10 15 20 25 30 350

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

SIF (MPa-m0.5)

Del

ta a

(um

)

K

MAX=15.4

KMAX

=19.2

KMAX

=30.7

KMAX

=30.7

KMAX

=30.7

KMAX

=33.9

KMAX

=33.9

KMAX

=33.9

Due to crack closure

[010]<001>crack

yyσ

7

Atomistic simulation

• Crack growth mechanism: microvoid nucleation and linkage ahead of the main crack tip

• Crack surface contact during the unloading path

8

Simulation results

Qualitative agreement with in-situ SEM testing Crack growth is not uniformly distributed within one loading cycle

9

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

The remaining question: how to efficiently track the CTOD variation of a growing crack?

•••

⋅−⋅= δδσσσ )()()( fHHa op

Analytical solution of CTOD variation under random loading

10

[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.

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”

11

Simulation results

1 1.1 1.2 1.3 1.4 1.5

x 104

0

2

4

6

8

10

12

14

16

18

20loading

Time

K(M

Pa*

m0.

5 )

1 1.1 1.2 1.3 1.4 1.5

x 104

2.031

2.032

2.033

2.034

2.035

2.036

2.037

2.038

2.039

2.04x 10

-3 crack length

Time N

a (m

)

50 100 1500

5

10

15loading

Time

K(M

Pa*

m0.

5 )

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

8.32

8.34

8.36

8.38

8.4

8.42

8.44

8.46

8.48x 10

-3 crack length

Time N

a (m

)

1000 1020 1040 1060 1080 11000

2

4

6

8

10

12

14

16

18loading

Time

K(M

Pa*

m0.

5 )

8900 8910 8920 8930 8940 8950 8960 8970 8980 89902.457

2.4575

2.458

2.4585

2.459

2.4595

2.46

2.4605

2.461

2.4615

2.462x 10

-3 crack length

Time N

a (m

)

Single overload Christmas Tree Spectrum Random Spectrum

Validation with in situ SEM testing

12

Constant amplitude loading R=0.1

0

1

2

3

4

5

6

7

0 10 20 30 40

CTO

D (u

m)

SIF (MPa-m0.5)

prediction testing data

Constant amplitude loading R=0.025

Validation with CT specimen testing

13

baselineLowratio

High ratio

20% 10%

14

sensor Critical spot

Concurrent structural dynamics and material fatigue damage prognosis

Mode extraction

2221

1211

φφφφ

21

11

21

11

δδ

φφ

=

Mode matrix

sensor data

unknown place

Extrapolation process for the unknown spots

Dynamic response reconstruction

0 0.5 1 1.5 2-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01IMF(1)--Mode 2

Time(sec)

Dis

plac

emen

t(m)

0 0.5 1 1.5 2-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-3 IMF(1)--Mode 3

Time(sec)

Dis

plac

emen

t(m)

0 0.5 1 1.5 2-4

-3

-2

-1

0

1

2

3

4x 10

-3 IMF(1)--Mode 4

Time(sec)

Dis

plac

emen

t(m)

1 1.1 1.2 1.3 1.4 1.5 1.6-3

-2

-1

0

1

2

3x 10

-3

Time(sec)

Dis

plac

emen

t(m)

Theoretical displacementExtrapolated displacement

0 0.5 1 1.5 2-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08IMF(1)--Mode 1

Time(sec)

Dis

plac

emen

t(m)

Research overview

Concurrent fatigue prognosis

0 1000 2000 3000 4000 5000-0.01

00.01

imf #

1

0 1000 2000 3000 4000 5000-202

x 10-3

imf #

2

0 1000 2000 3000 4000 5000-202

x 10-3

imf #

3

0 1000 2000 3000 4000 5000-505

x 10-4

imf #

4

aCCHHa ref

•••

−−= σ

λσλσσσ 21

2)()(

≤>

=0,00,1

)(xifxif

xH

Previous reversed

plastic zonea r

p

a Forward plastic zone during reloading

Δrp

a

Previous reversed

plastic zonea r

p

a Forward plastic zone during reloading

Δrp

a

)(tfkxxnxm =++•••

Structural dynamics

Fatigue crack growth

0 0.5 1 1.5 2-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Dis

plac

emen

t(m)

Time(sec)

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Kilocycle

Cra

ck le

ngth

(m)

15

Strain-based dynamic response reconstruction

Location of interestStrain gauge measurements

u

eElement

Measurement )()( teε

Modal responsesmite

i ...1),()( =η 1

)()(

)()()()( )()(

−

= u

iu

ei

ee

iu

i ttΦBΦBηη

Modal responsesmitu

i ...1),()( =η

Reconstructed )()( tuε

Decomposition(EMD)

1

Transformation2

Superposition3∑

=

≈mi

ei

e tt...1

)()( )()( ηε ∑=

≈mi

ui

u tt...1

)()( )()( ηε

16

Demonstration example - description

0 2 4 6 8 10-1

-0.5

0

0.5

1 x 10-3

Time (second)

Stra

in

(b)

(a)

(c)

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1 x 10-3

Time (second)

Stra

in

0 20 40 60 80 1000

0.5

1

1.5 x 10-4

Frequency (Hz)

Am

plitu

de

Fourier spectraIdentified frequencies

-1

0

1 x 10-3

Mod

al 1

-5

0

5 x 10-4

Mod

al 2

x 10-4

-5

-5

0

5 x 10-4

Mod

al 3

4 4.5 5 5.5 6-1

0

1

2 x 10-4

Mod

al 4

Time (second)

strain measurement

strain measurement

Fourier spectra of the measurement data

Four modal responses of the strain measurements data obtained using EMD

5m0.05m

1 2 3 4 5 6 7 8 9 10

Loc. 1 Loc. 2 Loc. 3

Strain measurements

hotspots

Random pressure

17

Demonstration example – verification of strain reconstruction

(a)

5 5.2 5.4 5.6 5.8 6-1

-0.5

0

0.5

1 x 10-3

Time (s)

Stra

in

TheoreticalReconstructed

( )

(c)

5 5.2 5.4 5.6 5.8 6-1

-0.5

0

0.5

1 x 10-3

Time (s)

Stra

in

TheoreticalReconstructed

5m0.05m

1 2 3 4 5 6 7 8 9 10

Loc. 1 Loc. 2 Loc. 3

Strain measurements

hotspots

5 5.2 5.4 5.6 5.8 6-1

-0.5

0

0.5

1 x 10-3

Time (s)

Stra

in

TheoreticalReconstructed

(b)

Results for Loc. 1

Results for Loc. 2 Results for Loc. 3

Random pressure

18

Demonstration example – verification of crack growth prediction

0 50000 100000 1500005

6

7

8

9

Time (s)

Cra

ck le

ngth

(mm

)

Theoretical 1% RMS noise 2% RMS noise 5% RMS noise10% RMS noise

0%noise

2%noise

5%noise

10%noise

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

Efficient probabilistic fatigue crack growth analysis

Key idea: matching the final life prediction of a realistic random loading to an equivalent constant loading

20

Equivalent random loading transformation – non-coupling case

mmi

iiii

n

mmi

iin

eq

gRgRp

gRg

NN

1

1

1

1

))0()(),((

))0()((

σσ

σσ

∆∆=

∆=∆

∑

∑

0

100

0 1 2Loading

Stre

ss

Δσeq

0

100

0 1 2Loading

Stre

ss

Δσ2 Δσ1

Same Life

10150

200250

300

-3-2

-10

1

0

0.02

0.04

0.06

0.08

Stress range (MPa)Stress ratio

f(x,y

)

0 1000 2000 3000 4000 50000.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0.021

0.022

Fatigue life (cycles )

Cra

ck L

engt

h (m

)

Random loading processEquivalent stress level

0 10 20 30 40 50-150

-100

-50

0

50

100

150

200

Faituge life (cycles)Lo

adin

g (M

Pa)

Key idea: using the time-based subcycle FCG model to calculate a correction factor distribution for the coupling effect

21

Equivalent random loading transformation – coupling case

eqeq σησ ∆=∆ *

0 10 20 30 40 50-100

-50

0

50

100

150

200

Fatigue life (cycles)

Load

ing

(MP

a)

Reversed plastic zoneForward plastic zone

Coupling zone

(rp)0L

rp rpv

a0L

a1.2

1.31.4

1.51.6

1.7

0

0.5

10.5

0.6

0.7

0.8

0.9

1

Overload ratio, xOccurrence probability, y

Fitti

ng c

urve

resu

lts, z

0.6 0.7 0.8 0.9 1 1.1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x)

empirical CDF

Fitting results

Random loading Time-based FCG model

Gaussian Process modeling

Probability Distribution of η

Key idea: calculate the remaining useful life (RUL) at a specified reliability level by solving the inverse reliability problem

22

Efficient probabilistic fatigue life prediction algorithm - iFORM

-2.5 -2 -1.5 -1 -0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Stress ratio

-2.5 -2 -1.5 -1 -0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Stress ratio

-2.5 -2 -1.5 -1 -0.5 0 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Stress ratio

Mechanism model

Input/quantified uncertainties

Output/propagated uncertainties

Probabilistic RUL Estimation

MPP

7 CT specimen testing under stationary random loading spectrum for FCG testing

23

Demonstration and validation of probabilistic prognosis

Time

Pmax,OL

Pmax

Pmin,

n 1 n1 n2 n3

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 CT-VI-6 n1=47, n2=0, n3=0 CT-VI-7 n1=12, n2=20, n3=15

2

4

6

8

10

12

14

16

0 10 20 30 40 50 60 70ThousandsFatigue life (cycle)

Cra

ck le

ngth

(mm

)

CT-VI-1

CT-VI-2

CT-VI-3

CT-VI-4

CT-VI-5

CT-VI-6

CT-VI-7

Median prediction

95% confidence bounds

Inverse FORM

Approach Computational time Point-by-point FCG simulation

(+MC simulation ) ~10 hours

Equivalent stress level (+MC simulation) 2309 seconds

Equivalent stress level (+Inverse FORM) 5 seconds

Equivalent Stress model + iFORM

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)

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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

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Acknowledgements

The research reported was supported by funds from

Air Force Office of Scientific Research (AFOSR) -Young Investigator Program (Contract No. FA9550-11-1-0025, Project Manager: Dr. David Stargel). The support is gratefully acknowledged.

Discussion and encouragement from Eric Tuegel, Reji John, and Ravi Penmetsa at AFRL.

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Thanks! Questions?