Small Crack Fatigue Growth and Detection Modeling with Uncertainty and Acoustic Emission Application

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Herbert Hawkins
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1 Small Crack Fatigue Growth and Detection Modeling with Uncertainty and Acoustic Emission Application Presentation at the ITISE 2016 Conference, Granada, Spain June 26 th 29 th, 2016

2 Outline Overview of Fatigue Crack and Acoustic Emission Experimental overview and examples of results Probabilistic Modeling Results Conclusions 2

3 Traditional Crack Growth/Detection Modeling Traditional empirical modeling of crack damage is typically used to model direct observable crack initiation and growth Recent advances in crack growth allows use of indirect precursors of crack such as acoustic emission (AE) for online monitoring Wide variety of crack growth and detection models, however these empirical models contain uncertainties: 1. Model/Data Uncertainty 2. Physical Variability 3. Model Error Identification of these uncertainties are necessary to improve crack growth/detection models 3

4 Fatigue Testing with AE Volts Rise time Counts Amplitud e Threshold Time Duration To preamplifier & frequency filter Surface disp. waves F Crack AE sensor F Specimen 4

Fatigue Testing with AE (Cont.) a b 8 1 1. Specimen 2.

Acoustic sensor 5. Extensometer 6. Optical microscope 7.

5 Fatigue Testing with AE (Cont.) a b Specimen 2. Clamped styrene-butadiene rubber 3. Wrapped neoprene strip 4. Acoustic sensor 5. Extensometer 6. Optical microscope 7. External light source 8. Testing grip SE 2016, July 27th-29th, Granada, Spain

6 Fatigue Testing and Test Example Sample Information: Ø Test CSFs Frequency = 2 Hz Load Ratio = 0.5 Min/Max Load = 6500/13000 N Units all in mm Al 7075-T651 material Ø Material CSFs Mean Grain Diameter WBL(69.7,1.61) Mean Inclusion Diameter WBL(10.4,1.68) Data 125 data collected (5 non-detections, 120 detections) 108 training data, 12 validation data da/dn = α 1 dc/dn α 2 Half circle notch measures 0.5 mm radius 6

Introduction of CSFs and Applications in Crack Growth

Test or material based input properties that have direct

Accounting minimize model uncertainties Force, Load

Crack length a Several growth and detection models were

7 Introduction of CSFs and Applications in Crack Growth Modeling Crack Shaping Factors (CSFs): Ø Definition: Test or material based input properties that have direct bearing on the shape, length, and growth of the crack Ø Accounting minimize model uncertainties Force, Load Ratio, and Load Frequency Grain size Inclusion size Crack length a Several growth and detection models were tested through probabilistic Bayesian parameter estimation procedures 7

8 Likelihood Function for Bayesian Analysis Find the model parameters with the following likelihood function, l(d=0,1; a i=1,, a n D, x i=1,, x n D, x j=1,, x m ND A, B, P FD )= i=1 n D [(1 P FD )POD(D=1 B, a i > a lth )f( a i A, x i )] j=1 m ND [1 (1 P FD ) a lth POD(D=1 B,a> a lth )f(a A, x j )da ] Terms: Ø A crack growth model parameters Ø B crack detection model parameters Ø P FD probability of false detection Ø a crack length at time point i or j Ø x crack shaping factor vector pertaining to crack length a i or a j Ø n D number of detection data points ( x i=1,, x n D ; a i=1,, a n D ) Ø m ND number of non-detection data points ( x j=1,, x m ND ; a j=1 =0,, a m ND =0) Ø POD probability of detection CDF Ø f crack propagation PDF Ø D detection state (D=1 detected/d=0 not detected) 8

9 Crack POD Model Choices Lognormal POD Model Ø POD(a ζ 0, ζ 1, a lth )= a lth a 1/(x a lth ) 2π ζ 1 2 exp { 1/2 [ ln (x a lth ) ζ 0 / ζ 1 ] 2 }dx ; B =[ ζ 0, ζ 1 ] Log-logistic POD Model Ø POD(a β 0, β 1, a lth )= exp [ β 0 + β 1 ln (a a lth ) ] /1+ exp [ β 0 + β 1 ln (a a lth ) ] ; B =[ β 0, β 1 ] Logistic POD Model Ø POD(a η 0, η 1, a lth )=1 1+ exp ( η 0 η 1 ) /1+ exp [ η 0 (a η 1 a lth )] ; B =[ η 0, η 1 ] Weibull POD Model Ø POD(a α 0, α 1, a lth )=1 exp [ ( a a lth / α 0 ) α 1 ] ; B =[ α 0, α 1 ] 9

10 Crack Growth Model Choices All crack growth models fit to the lognormal PDF distribution, Ø f(a A, x )= 1/aσ 2π exp [ 1/2 ( ln a ln g( A, x ) /σ ) 2 ] where σ is the standard deviation Log-Linear Growth Direct Model Ø ln a =b+m ln N Acoustic Emission Growth Indirect Model Ø a(n)={ αi(n)+β&linear@α I(N) β &power where I(N) is a function of AE amplitude and cumulative counts Gaussian Process Regression (GPR) Growth Model Ø a=g( x )=g([ CSF 1 & CSF 2 & & CSF Q ]) 10

11 More Details About the GPR Model a~nor[0,k([x], A )] where K(.) is the M M covariance matrix or kernel matrix that correlates a and [X] Kernel matrix are made up of kernel functions k( x i, x j, A ) taking two sets of CSF data x i and x j and the Gaussian crack length model parameters A k( x i, x j, A )= A 1 + q=1 Q A q+1 x i,q x j,q + A 2+2Q exp [ q=1 Q A q+q+1 ( x i,q x j,q ) 2 ] + A 4+2Q sin 1 A 3+2Q q=1 Q x i,q x j,q / ( A 3+2Q q=1 Q x i,q x j,q ) 2 ITISE + A 5+2Q 2016, July 27th-29th, δ i,j Granada, where Spain δ i,j is a 11 Dirac function

12 Crack Growth Model Choices Input/output relation of GPR Crack Length (µm) ! 441 = Growth Model Parameters Output: Crack Length (µm) Fatigue Cycles Min Load(N) Max Load (N) a1 CSF 1,1! a = CSF2,1 g 6500 $ A, = " " " g A a, 108 CSF108,1!!! Load Ratio CSF! ( 1,2 CSF A ) 2,2 X [ a ] g,[ ] " CSF!! Model Parameters 108,2 0.5 Input: Test and Material CSFs Frequency (Hz) 2 2! 2 Mean Grain Diameter α (µm) CSF 1,9! 69.7 CSF 1.61 # "! CSF! 69.7 Mean Grain Diameter β 2,9 108, Mean Inclusion Diameter α (µm) ! 10.4 Mean Inclusion Diameter β !

13 Fatigue Life Test Data Collection 100 magnification 200 magnification 13

14 Measurement Error Considerations Log-Linear-Based Crack Length AE-Based Crack Length Mean Measurement Error Model Error 51.9% 24.5% The model error between the true and estimated length is less for AE-Based crack length estimation than with the measured length estimation AE length measurement only overestimates the true length during early cycles 14

15 Results of Crack Growth Models Considered 15

16 Crack Growth/Detection Model Results Growth Model\POD Lognormal Logistic Log-Logistic Weibull AVERAGE Log-Linear NMSE AE GPR The GPR/Log-logistic pair has the smallest normalized mean square error (NMSE) of validation data Number of CSFs used understandably correlates to lower NMSE Ø GPR 9 CSFs Ø AE 3 CSFs Ø Log-linear 1 CSF 16

17 POD Model Results 17

18 Conclusions AE-Based model Ø Has a lower error value than that of measured length model error Ø Has improved over previous studies from 42% to 24.5% By implementing the GPR on 12 CSFs pairs the model error could be reduced by a large sum The number of CSFs used directly impact the fitness and realism of the crack length model. 18

19 THANK YOU! 19

An Acoustic Emission Approach to Assess Remaining Useful Life of Aging Structures under Fatigue Loading PI: Mohammad Modarres

An Acoustic Emission Approach to Assess Remaining Useful Life of Aging Structures under Fatigue Loading PI: Mohammad Modarres Outline Objective Motivation, acoustic emission (AE) background, approaches