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Procedía Engineering 10 (2011) 613-618
Probabilistic Strain Energy Life Assessment Model
R. Smitha, V. Ontiverosa, G. Paradeea, M. Modarresa* and P. Hoffman1
aUniversity of Maryland, College Park 20742, Maryland USA bNA VAIR Engineering Education & Research Partnerships, Patuxent River 20670, Maryland USA
Abstract
Under the "safe-life" methodology used by the United States Navy the service life of aircraft fleet are determined by monitoring the aircraft's usage and estimating the respective fatigue damage. The calculated fatigue life expenditure representing cumulative damage is then used to determine if it is possible for an aircraft or a fleet of aircrafts to reach the point of crack initiation and whether or not it should be retired. This research focuses on the development and initial application of a probabilistic strain-energy model to augment the empirical-based fatigue life expended approach. Experimental fatigue data obtained in this research is used to determine the relation between the number of cycles-to-failure and the cumulative total strain energy. A Bayesian framework for regression, including consideration of the model error is used to develop a probabilistic model of life that includes parameter uncertainties due to the limitation and scatter observed in the experimental data.
© 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of ICM11
Keywords: strain energy; probability; crack growth
1. Introduction
The safe-life methodology full-scale fatigue testing (FSFT) has been an established method of aircraft service life estimation that has been used by the United States Navy for the past thirty years. The FSFT procedure requires data acquisition from either in-flight or static simulation conditions. The program acquires data that includes the count and state of loads, strain, deflection, cracks, and crack growth. It then determines the rate of crack growth and subsequent failure based on the cumulative damage (i.e., Miner's Rule) which involves determining where cracks exist assuming a standard crack initiation size of 0.254 mm or larger. Reverse calculation is then used to obtain the number of cycles to crack initiation. In many cases, the calculated life by this approach can be much shorter than reality, which results in fleets of aircraft and rotorcraft being prematurely retired [1]. Additionally, this leads to purchases of new fleets to replace the retired ones, often prematurely based on the actual service life remaining. Based on the highly conservative methodology of the safe-life method, the University of Maryland is developing an extension of this method to estimate the service life of fleets of aircraft and rotorcraft. This methodology is based primarily on mechanistic and engineering methodologies including testing and probabilistic assessment of test data using Bayesian estimation. This paper will discuss the one aspect of the extended methodology involving
* Corresponding Author. Tel.: +1-301-405-5226; fax: +1-301-314-9601. E-mail address: modarres@umd.edu.
1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2011.04.102
development of a probabilistic strain energy-based structural life assessment model. In Section 2 the experimental setup will be defined as well as the analytical aspects of the data analysis. The results will be discussed in Section 3, as the estimated and actual crack growth results are compared and the parameters based on the Bayesian inference are defined.
Nomenclature
da/dN crack length growth per cycle
AK stress intensity factor
C material energy absorption capacity
m fatigue exponent
Wp plastic strain energy
We elastic strain energy
D damage
N test cycle
N cycle of crack initiation
AWtot cumulative strain energy
|j.p mean vector of parameters
Ep covariance matrix of parameters
2. Life Model Development
The purpose of the experiment is to propose and assess a model through which the remaining life of a structure can be estimated. The model proposed relies on the relationship between damage and absorbed strain energy. To develop and probabilistically estimate the parameters, a number of fatigue experiments have been performed.
2.1. Description of Experiments
This study focuses on the fatigue and crack initiation behavior of Al 7075-T651 samples. Seven samples were tested under constant tensile force at a loading ratio of zero. The design specifications for these samples are listed in
Table 1.
Table 1: Design Specifications for Al 7075-T651 Samples
Single Hole
Three Hole Samples
Length Width
27.7 mm
508 mm
38.1 mm
508 mm
Thickness
12.7 mm
12.7 mm
Hole Diameter
5.08 mm
6.35 mm
Hole Location The hole was located in the center of 152.4 mm from the top and
the coupon, 254 mm from the top and bottom of the coupon with 101.6 bottom mm of spacing between each hole
An MTS 810 uni-axial fatigue testing machine was used for all tests. Stress amplitudes ranging from 165 to 279 MPa were used. Each test was conducted at room temperature and a frequency of 2 Hz. Fig. 1 shows the MTS 810 testing machine and pictures of both the single-hole and three-hole specimens.
Periodically during the test, the areas around the hole were visually inspected for the presence of surface cracks. Upon the observation of a crack, rather than waiting for the sample to fail due to full fracture, the sample was removed from the MTS 810, and the edges were back-cut to a position just short of the crack length. Upon completion of back cutting, samples were then placed back into MTS 810 and pulled apart. This allowed for the fracture surface to be observed without requiring the extra time needed to allow the sample to fully fail due to fatigue, or contributing any additional uncertainty in determining the point of crack initiation by allowing the crack to propagate further.
Fig 1: (a) MTS 810 uni-axial fatigue testing machine, (b) single-hole specimen, (c) three-hole specimen
The tested samples were then analyzed using a Buehler ViewMet Inverted optical microscope to observe the crack initiation surface. Using image analysis software, the observed cracks were measured. An example of a crack surface used for measurement is shown in Fig. 2. These values were used in the Walker equation [2] to make an estimation of the number of cycles leading to crack initiation (assumed to be 0.254 mm).
da/dN = C (¿^f (1)
where, da/dN is the crack length growth per cycle, AK is the stress intensity factor, R is the stress ratio and C, m, and w are material constants. As mentioned above, in this initial work, each experiment was performed at a loading ratio of zero. The Walker equation is used in anticipation of future tests of non-zero stress ratios.
Fig 2: Crack surface measurements
2.2. Proposed Model and Data Analysis
Processing the data per specimen was done through an intricate set of MATLAB programs built upon a series of physical expressions. First, the hole-stress and strain per cycle was calculated from the MTS data using a combination of the Ramberg-Osgood Relationship and Neuber's Rule [3].
—/ w /—
Strain
Fig 3: Hysteresis Loop and the breakdown of total strain energy
Hysteresis loops, as shown in Fig. 3, were calculated for all test cycles at the hole. In each loop, the total dissipated strain energy was calculated by obtaining the area within the loop, which is Wp plastic strain energy, in addition to the elastic strain energy We, which is a sum the two adjoining areas outside of the loop[4].
Wtot = Wp+We=Wp+ W/ + We- (2)
The energy for the model is treated as cumulative, so the cumulative strain energy was calculated for each specimen. Damage, D, was calculated as a ratio of the number of test cycles N at a given point over the calculated number of cycles to crack initiation Ni.
D = (Ni- N/NO100 (3)
Finally, the proposed model correlated D and AWtot is represented as [5]:
mtot = CD~m or D = "7awtot/c
(4a-b)
To probabilistically estimate parameters C and m, a Bayesian parameter estimation procedure was developed. Equation (4a) was transformed into a general linear regression model to account for all of the tests within the same series [6]. Therefore,
log AWtot y = log Cj - nit log £>y + ey (5)
Here e is the residual deviation for specimen i at time ty which is modeled as a normal distribution (0, os). Using this form, the likelihood function becomes:
L()ogAWtotij ,log£>y |0/i,) = nL = l0gA"t0ti/'(^Ci'mil°g0i/) (6a-b)
Variable ©p represents the vector of parameters (i.e., C, m, and as) with mean and covariance matrix of parameters (^ Hp) [6]. Using a prior estimation of parameters from two sets of fatigue tests at low-cycle and high-cycle fatigue, to obtain the joint probability density functions of the parameters C, m, and as the Bayesian inference was solved using the program WinBUGS [7].
3. Experimental Results
3.1. Crack Growth Results
As discussed above, Walker's equation was used to estimate the number of cycles required to reach the point of crack initiation. For this study the point of crack initiation was assumed to have a length of 0.254 mm. General properties for the material constants C, m and w, as well as estimated stress intensity factors were determined using eFatigue calculators [8]. The results of this analysis are presented in Table 2. These results are then used to determine the damage amount by way of Equation (3).
Table 2: Experimental results and analysis.
Sample Stress Af Nf NRegion II NRegion I
Type (MPa) (mm) (experimental) (Estimated) (Calculated)
Three Hole 191 4.89 55487 28578 26909
Three Hole 165 0.68 64046 27223 36823
Single Hole 188 1.91 46687 25403 21284
Single Hole 248 1.98 10422 9948 474
Single Hole 248 4.26 14637 11188 3449
Single Hole 248 3.46 16800 10917 5883
Single Hole 279 1.53 12210 6227 5983
3.2. Parameter Estimation Results
Fig 4: (a-c) Marginal PDF of parameters, (d) parameter spread, (e-g) box and whiskers parameters, and (h) best fit model from program execution.
Fig. 4 contains the results of a run of the code, where 5000 iterations were performed and the following maximum life estimation applies. The following are the numeric results of this run of the code.
-0.99 0.0010 1.73 0.0000058
506.91 1.73 3467 0.0045
. 1.15 . .0.0000058 0.0045 0.00056 .
(7a-b)
Table 3: Mean, standard deviation (std), confidence intervals, and median of the parameters.
mean std 2.50% Median 97.50%
m -0.99 0.031 -1.05 -0.99 -0.93
C 506.91 58.88 403.83 502.95 636.51
a 1.15 0.02 1.11 1.15 1.20
The following mean life estimation model is formed based on the current output:
FLR = 1- 0.01(D) = 1- 0.01 (so^e)1'01 ~ 1 - (1-977 X 10"S)AWtote~E, £ = NOR(Q, aE) (8)
where FLR stands for the fraction of life remaining. 4. Future Work and Concluding Remarks
In the current and future work, efforts are being placed on further model refinement. Experimental results for non-zero stress rations of R = 0.1 and 0.4 are already under way. Additionally, testing scenarios that generate marker-bands, a technique used to create a specific pattern on the fracture surface and can be used to better determine the number of cycles after crack initiation will be used.
All new generated data will be used to update and improve the accuracy of the model presented in this paper. As more data is added, autocorrelation, an instance in which neighboring observations are non-independent, which results in biased ordinary least squared estimates are biased.
The process described in this paper was used to obtain a new life estimation model that is based on modern advances in reliability engineering and fatigue testing. As further tests are conducted the model described in Equation (4b) will be updated and adjusted until enough tests have been performed to fit the needs of the model.
Acknowledgements
The authors would like to thank David Rusk of the NAVAIR group for his considerable help References
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