method for monitoring fastener holes, and the crack propagation model is based upon Paris's Law. Simulated and experimental results are shown to assess the ...
A BAYESIAN APPROACH FOR ESTIMATING SIZES OF FATIGUE CRACKS NEAR FASTENER HOLES A. C. Cobb, J. E. Michaels and T. E. Michaels School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250
ABSTRACT. Ultrasonic in situ monitoring of metal alloys has been successfully demonstrated for determining the presence and size of fatigue damage within a stracture. Ultrasonic techniques, however, only provide an estimate of the state of the stracture at that time and do not predict the remaining fatigue life. On the other hand, a statistical crack propagation approach, which models the expected remaining life based on an assumed fatigue process, specimen geometry and material properties, allows for the fatigue life to be estimated. To maintain the safety of the stracture, this approach typically requires assuming a worst case initial flaw size. Presented here is a Bayesian estimation approach for incorporating both the measurement and modeling methodologies. An Extended Kalman Filter approximation is used to combine ultrasonic estimates of fatigue cracking with a crack propagation model. The measurement model is based upon recent work by the authors on a shear wave, angle-beam method for monitoring fastener holes, and the crack propagation model is based upon Paris's Law. Simulated and experimental results are shown to assess the performance of the estimation approach, where the resulting crack size determination is more accurate than either the ultrasonic method or the crack propagation model alone. Keywords: Ultrasonics, Fastener Hole Cracks, Stractural Health Monitoring, Kalman Filter PACS: 81.70.Cv, 43.35.Yb, 43.35.Zc
INTRODUCTION The sizing of fatigue cracks emanating from fastener holes is an important research area for Structural Health Monitoring (SHM) techniques. Fatigue crack initiation and growth from fastener holes has been studied extensively due to its importance in determining the remaining life of aerospace structures. Currently, the maintenance of many airframes is based on statistical modeling of crack growth within the specimen [1]. Also, techniques are described in the literature for the detection and sizing of defects based on ultrasonic measurements [2, 3]. However, the statistical approach does not actively interrogate the state of a particular structure and the measurement based techniques largely ignore the physics of fatigue crack propagation. The objective of this paper is to provide an estimate of the size of defects near fastener holes by combining fatigue crack growth laws with ultrasonic measurements using state estimation techniques. The result of this work demonstrates an improved estimate of crack sizes than would be possible using either of the established techniques alone.
CP975, Review of Quantitative Nondestructive Evaluation Vol. 27, ed. by D. O. Thompson and D. E. Chimenti © 2008 American Institute of Physics 978-0-7354-0494-6/08/$23.00
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BACKGROUND This section summarizes prior work performed by the authors pertaining to ultrasonic monitoring of fatigue crack growth near fastener holes [4, 5]. Experiments The specimens considered were 7075-T651 aluminum two-hole rectangular coupons of thickness 5.72 mm, length 305 mm, and width 47 mm. The two 4.82 mm diameter holes are in the center of the sample and are spaced 22 mm apart. The two holes act as stress risers for crack formation, and are far enough away from each other and the specimen edges that each can be considered independently. The samples were fatigued by repeating a purely tension fatigue spectrum of 2640 cycles, with the fatigue process being interrupted after every fatigue spectrum repetition (FSR) for ultrasonic measurements. The setup for ultrasonically monitoring the sample was to bond a pair of 10 MHz, 70 ° shear wave transducers in a through-transmission configuration on opposite sides of the hole with the direction of propagation perpendicular to the direction of crack growth. This transducer configuration maximizes the effect of the crack on the received ultrasonic signal. A sketch of the transducers and beam paths is shown in Figure 1. Energy Ratio Analysis The ultrasonic waveform feature used in this study is a ratio of signal energies from samples under tensile loading to their unloaded counterparts. The energy ratio is calculated based upon the signals recorded at no load and under a reference load, ^^,|,^^ ^ ERefLoadin) ^ ^^^^^ ^ ENoLoad(n)
ER'jn) E
^^^
R'{riiniual)
where E is the waveform energy versus cycles, n. The energy ratio response is plotted versus n, the number of cycles, where the indication of cracking is a sustained drop in the curve. ANALYSIS The ultimate goal of an SHM system is to track the remaining life of a structure in realtime and thereby reduce the chance of failure before maintenance can be performed. The remaining life is directly related to damage within the structure, specifically the presence and size of cracks emanating from fastener holes. The crack depth(s), considered to be representative of the state of the structure, should be estimated with a high degree of accuracy. State estimation techniques were employed to combine a measurement model, which maps the ultrasonic energy ratio to the maximum crack depth, with a system model, which propagates a crack according to crack growth laws. The state estimation framework tracks the depth of detected cracks, where a crack is considered to be detected if the energy ratio drops below a threshold.
Single V path
Double V path
FIGURE 1. Transducer mounting locations and ultrasonic beam paths.
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Extended Kalman Filter There are inherent inaccuracies in any measurement model or system model. The Bayesian estimation approach, which optimally corrects for these errors using the statistics of the models, is a classical method for reducing these inaccuracies, but it is generally impossible to implement because all needed statistics of the models are typically unknown [6]. For a practical implementation, assumptions have to be made about the models. For example, the Kalman filter implementation requires that the models be linear. The state estimation approach for this research is to use an Extended Kalman Filter (EKF) to combine ultrasonic measurements with a fatigue crack growth (FCG) power law. The EKF is a modification of the basic Kalman filter to allow for nonlinear measurement and system models by local linearizations about the current state estimate [6]. Compared to the original Kalman filter, the state update function. A, and observation (or measurement) function, C, may depend on x{n), the state at time n, x(n + 1) = A(x(n),n)
+ w(n),
y(n) = C^(x(n),n)
+ v(n).
(2)
The noise functions, w and v, are required to be uncorrelated, zero-mean Gaussian. To linearize these models, both are expanded via a Taylor series, where first order terms are used for the approximation. The procedure can be summarized in two steps. The prediction step, given as, x{n\n — I) = A{x{n — l\n — l),n — I),
(3)
uses the EKF response from the previous time step, x{n— l\n— 1), and the provided system model to predict where the state would be given no additional measurements. The update step, which corrects this prediction with an additional measurement, y{n), is given by, y{n) = C{x{n\n — 1)),
x{n\n) = x{n\n — 1) + K{n) [y{n) — y{n)],
(4)
where y{n) is the estimate of the measurement using x{n\n — 1). The Kalman gain term, K{n), incorporates the underlying statistics of the two models to optimally weight the correction from the new measurement, specifically incorporating the linearized models and their associated noises. A more complete description of the EKF is given in [6], where it is shown that the EKF implementation requires not only the system and measurement models, A{x) and C{x) respectively, but also their associated noise covariances, Qu, and Q^. Measurement Model An ultrasonic-crack interaction model was developed for understanding the expected energy loss as a function of crack geometry, and is described in more detail in [5]. To summarize, the approach is to assume a particular beam profile along the surface of the fastener hole and that any cracks present would block the beam over the associated crack area. The beam profile is modeled with an energy density function, given by e, {x,y), where i denotes a side of the hole. The {x, y) coordinates define a 2-dimensional slice through the sample. To obtain the energy ratio response associated with a given crack geometry, two energy integrals are calculated. The first is the received energy associated with an unloaded specimen, which is taken to be the maximum energy possible. The underlying assumption is that residual stresses completely close any cracks at zero load. This unloaded response at the receiver is given by, Eunioaded= j
ei{x,y)dA+
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e2{x,y)dA,
(5)
where Gi and G2 define the specimen geometry on the two sides of a single hole. The second energy integral corresponds to the loaded specimen, where it is assumed that the applied stress fully opens any cracks present. The energy that arrives at the receive transducer is determined by reducing the unloaded energy, Eunioaded, by the energy blocked by the cracks. El,
ei(x,y)dA-
= E,, Hi
e2(x,y)dA,
(6)
H2
where Hi and H2 define the crack areas on each side of the fastener hole. Finally, synthetic energy ratio values are determined by calculating the ratio of the loaded energy to the unloaded energy. Thus, given a realistic energy density ei{x, y), the effect of cracking on the received ultrasonic energy can be modeled. The ultrasonic energy density, ei{x,y), was approximated as a Gaussian distribution along the surface of the hole (x direction) and an exponential decay away from the hole surface (j/ direction) [5], „
-(x-h/2)^
ei{x,y) = e-i^^e '^'^'^ , (7) with i defining a side of a single hole. The parameters /3 and a^ are taken from [5], and are 3.44 mm"^ and 3.8 mm, respectively. To develop the required measurement model, C{x), and the associated measurement model noise, Qv, for the EKF framework, a Monte Carlo distribution of crack profiles was generated and energy ratio curves were calculated using this ultrasonic-crack interaction model; identical cracking on both sides of the hole was assumed. For each iteration of the Monte Carlo process, a crack profile was created for one side of the hole. The crack profile was formed through the union of a random number of half ellipses, where the aspect ratios, depths, and positions were randomly generated using probability distributions chosen such that the resulting crack profiles were somewhat realistic. A total of 200,000 crack profiles were generated and processed using the ultrasonic interaction model to create a distribution of the mean energy ratio response as a function of crack depth, as shown in Figure 2(a). This curve fit is the measurement model, C{x). Similarly, the measurement model standard deviation versus crack depth was computed from this data, and is shown in Figure 2b. Crack Propagation Model The final modeling aspect is determining an appropriate model and associated standard deviation for fatigue crack propagation. It is desirable that this model be differentiable and based on established mechanics theories. As such, the original Paris's Law is used as the system model, which relates stress intensity calculations to material properties to determine
0
0.5
1
1.5
2
2.5
Max Crack Depth (mm)
(a) Curve fit to energy ratio mean response
0
1
2 Depth (mm)
3
(b) Standard Deviation of curve fit
FIGURE 2. Measurement model and associated noise statistics derived from Monte Carlo simulations.
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incremental fatigue crack growth [7]. Paris's Law is given as, dx (8) = C{/\Kf dN Here x is the crack depth, C and m are material properties, and /^K, the stress intensity range, is, Aii' = i^AS-^/^, (9) where x is the crack length, /S.S is the applied stress range for that cycle and F is the sample geometry correction factor. The applied stresses are given by the spectrum fatigue loading used during the fatigue process. For the work presented here, the F parameter is taken to be. F{x) =F =
1 + 1.1243(1-
(10)
which assumes a through thickness crack from a circular notch of radius r [7]. An example crack growth curve is shown in Figure 3(a). The system model, ^(a;), is then defined as. Xk+i = A(xk) = Xk + 'Y^ C(AKi]
(11)
where the summation is over an entire fatigue spectrum repetition. The standard deviation of this system model cannot be directly calculated given that Paris's Law is deterministic. By assuming that the only variation in fatigue crack growth curves comes from the initial crack size, the statistics of the system model are approximated via a Monte Carlo simulation using a Gaussian distribution for the initial crack sizes. The mean and standard deviation were defined to match experimental observations, causing sample failure after 20 FSR on average and a variation of approximately ±4 FSR before failure. The crack depth standard deviation is calculated after every FSR, as shown in Figure 3(b). RESULTS This section summarizes the performance of the EKF framework for estimating crack sizes. The majority of the results deal with tracking damage from simulated energy ratio curves; experimental results are reported from two specimens.
5 Fatigue Spectrum Repetitions
(a) Example crack growth curve
10 15 20 25 Fatigue Spectrum Repetitions
30
(b) Standard deviation of crack depths
FIGURE 3. System model and associated noise statistics.
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; 0.8; 0.6•0.4n \ 0.25
10 15 Fatigue Spectrum Repetitions (FSR)
FIGURE 4. Example simulated energy ratio curve (dashed line) and associated crack depths (solid line).
Numerical Results The results presented here track fatigue crack depth from synthetic energy ratio curves. The curves are created by first generating a fatigue crack growth curve using Paris's law based upon an assumed crack position and aspect ratio. The crack shapes are then applied to the ultrasonic interaction model to generate a simulated energy ratio response at each measurement point, and Gaussian noise is added to this energy ratio curve. Figure 4 illustrates a typical simulated energy ratio curve along with the underlying crack depths. Concerning the position and aspect ratio assumptions, the original system model assumed through-thickness cracking; however, the reported crack depth from the Paris's Law model is interpreted to be the maximum crack depth, allowing for elliptical crack profiles to be considered. An example EKF response is shown in Figure 5. For the two system model response curves shown, a fatigue crack was propagated based on two different initial crack sizes. It is apparent that the initial crack size assumption determines how closely the model matches the "real" crack size. A simulated measurement model response is also provided, which gives a reasonable approximation of the real crack size, but is limited to approximately 2 mm in depth. Finally, the reported state estimation response from the EKF closely follows the simulated real crack growth curve, indicating an overall improvement in performance using state estimation. To assess the statistical performance of the state estimation framework, a series of simulated energy ratio curves were created with different assumptions for crack aspect ratio, crack position, and added electronic noise. The metric used to compare the results was a weighted root mean squared error {EWRMSE), defined as. SiWRMSE
= Vniean(w;fc(2;fc - Xk)
(12)
where Xk is the real crack depth, Xk is the estimate from the EKF, and Wk is the weight function, with the index k going from crack detection, ko, to the final measurement, kf. The
—0— Reai Cracl< Size (mm) —e— Kaiman Fiiter Approximation (mm) IVleasurement IVlodei Response (mm^ —
System IVlodei (a^ =0.0015 mm)
—X-System Model (a =0.0025 mm)
0» ^ • » - ^ » | . » ^ » (»^ » ^1^ » • Fatigue Spectrum Repetitions
FIGURE 5. Example Kaiman filter response using a simulated energy ratio curve.
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TABLE 1. EwRMSE Response with varying noise levels and aspect ratios (centered crack)
ER Noise STD 0.010 0.020 0.030 0.075 0.100
2:1 0.51 0.50 0.49 0.57 0.63
Aspect 4:1 0.23 0.24 0.33 0.38 0.38
Ratio 6:1 0.10 0.08 0.17 0.13 0.27
TABLE 2. EwRMSE Response with varying crack positions and aspect ratios (noise a = 0.02)
8:1 0.10 0.08 0.15 0.09 0.19
Crack Position Oh 1/4/i 1/2 h 3/4/1 Ih
2:1 0.75 0.52 0.50 0.53 0.76
Aspect 4:1 0.60 0.26 0.24 0.27 0.62
Ratio 6:1 0.52 0.15 0.08 0.14 0.51
8:1 0.43 0.08 0.08 0.09 0.44
weight function is linear. kf-k (13) kf - ko to more heavily weight the small crack regime immediately after deteciton. The median value for the EwRMSE metric is reported from 50 iterations. This metric allows a relative measure of performance between the different cases considered. The results of Table 1 show the effect of varying the added noise and crack aspect ratio with the crack position fixed at the center of the thickness. The results show that for reasonable amounts of measurement noise added to the energy ratio curve (u < 0.03), the EKF response was effectively unchanged. For noise exceeding a = 0.03, there was an increase in the EWRMSE response, indicating a performance degradation. Also, regardless of noise level, as the aspect ratio increases, the crack shape gets closer to the original assumption of a through thickness crack, yielding better performance overall. The second set of reported results, shown in Table 2, fixes the measurement noise at a = 0.02 and varies the crack position and aspect ratio. As the crack position deviates from the center of the thickness (1/2 h), the performance was degraded. This degradation is expected given that the measurement model assumes that energy is concentrated in the center of the thickness. Wk
Experimental Results Experimentally generated energy ratio curves were processed using the EKF framework. The material fatigued was 7075-T651, the same as for the derivation of the system model. Specimens were selected with cracking approximately symmetrical on both sides of the hole, as this assumption was made for development of the state estimation models. Results from two fatigue tests are reported as shown in Figure 6. Figure 6(a) illustrates small crack
- EKF Approximation (mm) IVleasured Cracl< Size i
-EKF Approximation (mm) IVleasured Cracl< Size
0.6 -0.4
02 0
0
5 10 15 20 Fatigue Spectrum Repetition (FSR)
(a) Specimen S4-0031, Hole 2
5 10 15 20 25 Fatigue Spectrum Repetition (FSR)
(b) Specimen S4-0191, Hole 2
FIGURE 6. Experimental results from two fatigue samples.
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performance (0.25 mm crack depth) and Figure 6(b) illustrates large crack performance (3 mm crack depth). For both cases, a good estimate of the final crack size was obtained. As opposed to the simulated results, only the final crack dimension was known, so it was not possible to quantitatively assess the EKF performance prior to the final measurement. CONCLUSIONS 1. The state estimation approach considered provides crack size estimates using an ultrasonic measurement model combined with a crack propagation law. For the numerical results shown here, crack sizes are more accurately estimated than with either the fatigue crack growth approach or the measurement based approach alone. 2. The state estimator performed well in the presence of reasonable additive noise for the simulated energy ratio curves. For larger noise levels, the state estimator was functional but the performance was degraded. 3. The Extended Kalman Filter performed better for cracks with larger aspect ratios and for cracking centered with respect to the specimen thickness. 4. The state estimation response for the two experimental curves considered yielded close estimates of the actual crack sizes at the final measurement point. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office, "Structural Integrity Prognosis System" Program, contract No. HROO11-04-0003 to Northrop Grumman Corporation, Integrated Systems. This paper is approved for public release, distribution unlimited.
REFERENCES 1. R. B. Thompson and D. O. Thompson, "Ultrasonics in nondestructive evaluation," in Proceedings of the IEEE, vol. 73, pp. 1716-1755, IEEE, 1985. 2. J. L. Rose and H. Schlemm, "Equivalent flaw size measurements and characterization analysis," Materials Evaluation, vol. 34, no. 1, pp. 1-8, 1976. 3. P. B. Nagy, M. Blodgett, and M. Golis, "Weep hole inspection by circumferential creeping waves," NDT&E International, vol. 27, no. 3, 1994. 4. B. Mi, J. E. Michaels, and T. E. Michaels, "An ultrasonic method for dynamic monitoring of fatigue crack initiation and growth," /. Acoust Soc. Am., vol. 119, no. 1, 2006. 5. J. E. Michaels, T. E. Michaels, and B. Mi, "An ultrasonic angle beam method for in situ sizing of fastener hole cracks," Journal ofNondestructive Evaluation, vol. 25, no. 1, pp. 215, 2006. 6. T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing. Pearson Education, first ed., 2000. 7. N. E. Dowling, Mechanical Behavior of Materials. New Jersey: Prentice-Hall, Inc., tenth ed., 1993.
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