CC BY-NC-ND
0Vol.18 No. 4
CH INESE JO L RN AL OF AERON A L TICS
November 2005
Corrosion Fatigue Life Prediction of Aircraft Structure Based on Fuzzy Reliability Approach
TAN Xiao-ming1, 2, CHEN Yue-liang1, 3, JIN Ping1
(1. Naval Aeronautical Engineering Institute Qingdao Branch, Qingdao 266041, China)
(2. Postgraduate T eam, N aval Aeronautical Engineering Institute, Yantai 264001, China)
(3. Aeronautical Academy, Northwestern Poly tech ni cal University, Xi an 710071, China)
Abstract: Material performance of LY12CZ aluminum is greatly degraded because of corrosion and corrosion fatigue, which severely affect the integrity and safety of aircraft structure, especially those of the navy aircraft structure. The corrosion and corrosion fatigue failure process of aircraft structure are directly concerned w it h many factors, such as load, material characteristics, corrosive environment and so on. The damage mechanism is very complicated, and there are both randomness andfuzziness in the failure process. With consideration of the limitation of those conventional probabilistic approaches for prediction of corrosion fatigue life of aircraft structure at present, and basal on the operational load spectrum obtained through investigating service status of t he aircraft in naval aviation force, a fuzzy reliability approach is proposed, which is more reasonable and close' to the fact. The effects of the pit aspect ratio, t he crack aspect ratio and all fuzzy factors on corrosion fatigue life of aircraft structure are discussed. The results demonstrate that the approach can be applied to predict the corrosion fatigue life of aircraft strucr t ur e.
Keywords: aircraft structure; corrosion; life prediction; fuzzy reliability; corrosion fatigue №) , 2005, 18(4): 346 - 351.
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Corrosion and corrosion fatigue are recognized as significant damage mechanisms to the navy aircraft structures. With extension of operation life, the problem is becoming increasingly important and has received more and more attention of the aeronautic researchers. The corrosion and corrosion fatigue failure process of aircraft structure are directly concerned with many combined factors, such as load, material characteristics, corrosive envirorr ment and so on. T he damage mechanism is very complicated, and there are both randomness and Received date: 2005 01-21; Revision received date: 2005 07 06
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fuzziness in the failure process.
Harlow and Wei[1] proposed firstly that corrosion fatigue life of aircraft structure should be composed of three stages: crack nucleation life, surface crack growth life and through crack growth life, and a probabilistic model was established. Vasude van'2] thought that the fatigue life should be the sum of four stages. And a seven-stage probabilistic model was proposed by Pan Shi'3,4].
However, there are two limitations of these models. Firstly, they are all based on constant am-
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plitude spectrum, without consideration of the operational load spectrum of aircraft structure. More important, the models are all based on conventional probabilistic theory, and only the random factors are analyzed, without consideration of the fuzzy factors that are existent in fact.
1 Analytical Model
The corrosion and corrosion fatigue damage process are assumed to begin with the nucleation of localized corrosion pit and subsequent corrosion fatigue growth and failure at last. Suppose the total corrosion fatigue life is the sum of four stages:
tf = t 1 + t2 + t3 + t 4 ( 1)
where 11 is the time for pit nucleation; 12 is the time for pit growth; 13 is the time for short crack growth to long crack; 14 is the time for long crack growth till failure.
The object of the proposed analytical model is to compute the failure probability at some specified time t, the cumulative distribution function (CDF) of the corrosion fatigue life can be expressed as:
P f(t) = P(t f- t < 0) (2)
1.1 Pit nucleation stage
After the failure of the corrosion prevention coating on the skin, the rivet hole is bare, directly exposed to corrosive medium and easily corroded. This stage includes the failure process of coating, subsequently the electrochemical corrosion process of aluminum alloy and the nucleation of a corrosion pit. The time of this stage depends on factors such as load, corrosion environment, and material properties of aluminum alloy, manufacture technology and so on. In addition, the damage mechanism is very complicated, and not well understood until now.
Only after the prevention coating has failed, is the corrosion environment able to corrode the material matrix. Assume the time for pit nucleation 11 is a Weibull random variable. According to Refs. [5, 6] the operational life of the prevention coating is
about -3"5 years China Academic Journal Electronic
1.2 Pit growth stage
The shape of the pit is assumed to be half of an ellipsoid ( Fig. 1) . For convenience, the aspect ratio is defined as
♦ = i (3)
where a and b denote half-length of the major and minor axes. The pit can continue to grow deeply into material matrix in corrosion environment. Its aspect ratio 1, and it is a random parameter. According to Ref. [7], the aspect ratio is in [ 1, 4. 72] and its mean value is 1. 5.
Fig. 1 Sketch of semi ellipsoidal pit
Applying Faraday' s law,
dV _ d( 2 Jlq&7 3) _ d t _ d t ~ nF p
By integration of Eq. ( 4) , the time for pit growth is obtained by
2 ^nF p
f 3 33I
ci - ao|
where aci is the critical pit size leading to short crack initiation, a0 is the initial pit size.
The transition from pit to short crack growth occurs when the equivalent stress intensity factor range !K for the pit increases to the threshold driving force !Kth[ 1]. For illustration purpose, it is assumed that the pit is the semi"elliptical surface crack in a semi"infinite body. According to the stress intensity factor handbook, the stress intensity factor range !K of the deepest point in the pit is expressed as
1. 1 K t AQ Jïq
( 1 + 1. 464 & 1 •65)172
The critical pit size can be found to be
( 1 + 1. 464 4>- 1 •65 )
a ci _
1. 1K t AQm
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348 •
TAN Xiacr ming, CHEN Yue liang, JIN Ping
where Kt is the stress concentration factor as a result of the rivet hole, AOmax is the maximal stress range in the load spectrum. 1.3 Short crack growth stage
The Walker' s formula is adopted to compute the crack growth rate, with consideration of the stress ratio
d a dN
= Csc[( 1 - R)m-1 AKyn
where R is the stress ratio, Csc is the short crack growth coefficient of LY12CZ aluminum alloy in corrosion environment, n sc is the short crack growth exponent and is assumed to be constant. Taking into account N = ft and combining Eq. (6) and Eq. ( 8) , the time for short crack growth is found t o be
2( 1 + 1. 4644>- 165)
fC sc ( 2 - nsc)
2 n sc - ( J"^ 2-n sc/
[ 1. 1( 1 - R)m- 1Kt where ath is the critical crack size for the transition from the short crack to the long crack, f is the frequency of the load spectrum block. At first, the operational loading spectrum is obtained through investigating operation status of the aircraft in the naval aviation. Secondly, the load is averaged in every month and the frequency f is obtained and its unit is cycle/month. Then, the actual load of the aircraft structure is gained through finite element analysis.
1. 4 Long crack growth stage
As for long crack, the stress intensity factor range is given by
AK = K t AO jTh (10)
According to Walker' s formula, the long crack g row t h rate is expressed is da
= Cic[( 1 - R)m- 1 AKynic ( 11)
T he time fcr long crack growth is expressed as
I a th ) 2-n fc7
fCic(2- nic)[( 1 - R)m- 1KtAoyipc
where af is the critical crack length.
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2 Analyses of Fuzzy Factors
2.1 The fuzziness of the structural failure criterion
T he corrosion and corrosion fatigue failure are caused by accumulative damages that degrade the material performance. The process from " good condition" to " failure condition" is a gradual process from " quantitative change" to " qualitative change" . It is neither " perfectly good" nor " com -pletely disabled" in the transition processes. Therefore, there is a fuzzy condition.
Ref. [1, 3, 4] presented a critical size of crack as the failure criterion for the structure, which was 6 mm. It implied that a crack of a =6 mm would not cause failure, whereas a crack of a = 6. 01 mm would cause failure. However, there is no substantive difference between 6 mm and 6. 01 mm. The process from the safe condition to the failure condition is gradual. Obviously, this rigid failure criterion is not reasonable.
The semi"trapezoidal membership function of the critical crack length af is adopted, as shown in Eq. (13) and Fig. 2,
Fig. 2 Semi"trapezoidal membership function
#a f (x) = i
a 2 - a 1 1
a 1 < x ^ a 2 ( 13)
x > a 2
where a 1=6, and a 2is determined based on experience, a2= ( 1.05-1. 3) a 1. Herein a2= 6. 3. 2. 2 The fuzziness of the transition criterion from short crack to long crack
According to Ref. [ 3], the critical size a th of
the transition of the short crack to the long crack is Publishing House. All rights reserved; http :/nvww. criki.net
1. 0 mm in engineering field. It is experiential, and there is lots of fuzziness obviously. The semi" trapezoidal membership function is adopted, as shown in Eq. ( 14) and Fig. 3,
(x)= \
x - a i
a i < x ^ a 2 ( 14) x > a 2
Where a2= 1. 0, a 1= (0. 77-0. 95) a2 Herein a 1= 0.77.
ath/mm
Fig. 3 Semi-trapezoidal membership function
2. 3 The fuzziness of the transition criterion from pit to short crack
According to Eq. ( 7), the critical crack size of the transition criterion from pit to short crack is determined by the random variable !K th, which is the material parameter and obtained through test. Because the test conditions are different, the specimen is also different from the actual structure, and there is the subjective factor, the threshold driving force !K th is a typical fuzzy random variable. In this paper, the fuzziness of the transition criterion from pit to short crack is described through !K th.
According to Ref.[ 8], a normal membership function is adopted, as shown in Eq.( 15) and Fig.4,
#ak (x) = \
(x - a 1
exp - 2 x < a 1
i_ a 2
^ 1 x % a 1
3 Determination of These Variables
3.1 Deterministic variables
The material of the skin is a LY12CZ aluminum alloy, the density P= 2700 kg/m3; the activation energy !H = 50 kJ/ mol; Faraday' s constant F = 96 514 C/mol; the universal constant R =8. 314 J/mol- K; the stress concentration factor of the rivet hole K t= 3; in the corrosion environment, the crack growth exponent nsc = nlc = 3. 14[9], and the impact exponent of stress ratio m = 0. 66[9]; the environment temperature T = 293 K .
3.2 Random variables
It is assumed that Weibull probability density function ( PDF) is chosen for these random variables, because of its much applicability. The randomness is depicted by the different shape parameter a, the minimum value parameter % and the scale parameter &. The three-parameter Weibull PDF is given by
f(x) =
/ N. f N
x - % 0- 1 x- % 0
. & . exp _ & V J _
© 1994F2f)40 (N°rnaa^^meîmben!i1c^(uncniân Electronic
The Weibull parameters of pitting current constant IP0 and initial pit size a0 are given according to Ref. [ 1]. Those of short crack growth coefficient Csc and long crack growth coefficient Cb are given according to Refs. [ 3, 9]. Those of the aspect ratio 0 are given according to Ref. [ 7].
4 Results and Discussion
According to these equations herein before, the corrosion fatigue fuzzy reliability life t f can be given by
tf= tf(IP0, a0, Csc, Clc, fy 11, AK th, «th, !f)
where IP0, a0, Csc, Clc, ^ and 11 are random variables, AK th, ! th and af are fuzzy variables. Com -bin in g these equations above, t f can be found to be
- exp( AH/8. 314 T)
U = 11 + T a2 •
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350 •
TAN Xiao- ming, CHEN Yue- liang, JIN Ping
[ 0. 57 Y3( AKth/ Aqmax) 6 - 2. 34 & 104a3 • 0. 0068[7. 5 Y( AKth/ AQmax)- 114 - Y157(!th)- 0 57] fCsc[( 1 - R)m1 Aoy3'14
Q 0092[(!th)
- 0. 57
✓ ~ r 0. 57-¡
(afj ]
fCfc[( 1 - R)m- 1 AO/3 w h ere Y = 1+ 1.464* . It is able to compute the failure probability at some specified time t by use of Monet-Carlo simulation method. 4.1 Effects of aspect ratio of pit
The effects of the pit shape and the short crack shape on corrosion fatigue life are shown in Fig. 5. The aspect ratio 0 is a random variables in Curve A , and it is in [ 1, 4. 72] according to Ref. [7]; in Curve B the pit is semispherical and the short crack is semicircular[1,3], 0= 1 . Obviously, the probabilistic life is decreased greatly by the randomness of the aspect ratio.
; time/months
Fig. 5 Effects of the aspect ratio on corrosion fatigue life
When the reliability is 0. 9999, 0. 999 and 0. 99, the corrosion fatigue lives are 338. 2, 427. 7 and 543. 9 months respectively in curve A , and they are 379. 4, 477. 3 and 603. 5 months respectively in Curve B. Evidently, the effect of the aspect ratio on the reliability life is great. In Refs. [1, 3], it was supposed that the pit was semi-sphere and the short crack was semicircle, the aspect ratio 0= 1. There is much limitation and the corresponding result may be dangerous. 4. 2 Effects of fuzzy factors
The effects of fuzzy factors on corrosion life are shown in Fig. 6, there is no consideration of fuzzy factor in Case A , the fuzzy factors of !f, !th, AK th and the combining fuzzy factor are discussed respectively in Case B, Case C, Case D and Case E. Th
Fig. 6 Effects of fuzzy factors on corrosion fatigue life
Curve B and Curve C, and so it does between D and Curve E.
The corrosion fatigue lives under different reliabilities are shown in Tabel 1. It is obvious that th e corrosion fat ig ue life is increased b y all of t he fuzzy factors, especially b y t he fuzzy factor of
Table 1 Corrosion fatigue lives under different reliabilities
Reliability Corrosion fatigue life/months
Case a Case b Case c Case d Case e
0. 9999 338.2 339. 3 338. 8 393. 8 395. 5
0. 999 427.7 429. 0 427.9 500. 9 502. 4
0. 99 543.9 545. 4 543.5 640. 5 641. 5
0. 9 697.9 699. 5 696. 3 825. 7 825. 8
When the reliabilities are 0. 9999, 0. 999 and 0.99, the fuzzy reliability lives are 33, 41. 9 and 53. 5 years respectively in Case E, which agree with the actually operational lifes of the aircraft structure. So, the validity and feasibility of the approach presented in this paper can be demonstrated to some ex tent .
5 Conclusions
Based on fuzzy mathematics theory, a fuzzy reliability model to predict the corrosion fatigue life has been established. By analyzing, a few of practical conclusions are gained.
(1) In this paper, the aspect ratio 0 is assumed to be a random variable, the analytical model is closer to the fact and the calculating results are safer.
(2) Not only the effects of random factors but also those of fuzzy factors on corrosion fatigue life,
, There is no evident difference among Curve A , can be taken into consideration by using the fuzzy © 1994-2010 China Academic Journal Electronic Publishing House. All rights reserved/ nttp://www.cnki.net
reliability method adopted in this paper, which is more reasonable than the conventional reliability approach.
(3) In view of the analysis results, the life is increased by all the fuzzy factors to different extent, especially the fuzzy parameter AKth. Therefore t hat , in order to predict t he reliability life of aircraft structure, the threshold driving force AK th must be measured precisely.
(4) The calculating model is established based on the actually operational load spectrum and not on constant amplitude spectrum as shown in Ref.
[ 1, 3] . So, the fuzzy reliability approach can be used t o predict t he corrosion fatigue life of aircraft structure.
References
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[2] Vasudevan A K, Sadananda K. Environmental effects on fatigue crack initiation and growth[ A] . In: RT O meeting proceedings 18[ C] . France: NATO Research and Technology Organization, 1999. 144- 156.
[3] ShiP, Mahadevan S. Probabilistic estimation of pitting corrosion fatigue life[ R]. AIAA- 2000-1644, 2000.
[4] ShiP, Mahadevan S. Aircraft structure reliability under corrosion fatigue[ R]. AIAA-2001-1377, 2001.
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^IMW^ [J]. IS^ft 2002, 23( 3): 249- 251. Chen Y L, LU G Z, Duan C M. A probability model for the corrosion damage of aircraft structuse in service environment [J]. Acta Aeronutica et Astronatica Sinica, 2002, 23(3): 249- 251. (in Chinese)
[7] Harlow D G, WeiRP. A probability model for the growth of corrosion pits in aluminum alloys induced by constituent parti cles[ J] . Engineering Fracture Mechanics, 1998, 59( 3): 305- 325.
[8] Rao S S, Sawyer J P. Fuzzy finite element approach for the analysis of imprecisely defined systems[J]. AIAA Journal, 1995, 33( 12): 2364- 2370.
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^)[M]. ^M: 1996. 227- 255.
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Biographies:
AN Xiao- ming Born in 1975, he received a B. Sc, and an M. E. from iavy Aeronautical Engineering Institute In 1999 and 2002. He is now a Ph. D. candidate. His research interests include corrosion fatigue of aircraft structure and H life reliability. Tel: 0532- 88033164.
CHEN Yue liang Born in 1962, a professor of Naval Aercr nautical Engineering Academy Qingdao Branch. He is now a Ph. D. candidate of Northw estern Poly technical U niversity. His research interests include corrosion fatigue and calendar life of aircraft structure.
JIN Ping Born in 1959, a professor of Naval Aeronautical Engineering Academy Qingdao Branch. He is now a Ph. D candidate of Beijing University of Aeronautics and Astronautics. He has won more than ten times of award for Promotion of Science and Technology from the state and the Army. His research interests include fatigue life and reliability of aircraft structure. Tel: 0532 88033161.
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