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Total fatigue life prediction for Ti-alloys airframe structure based on durability and damage-tolerant design concept
Materials and Design 31 (2010) 4329–4335
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Total fatigue life prediction for Ti-alloys airframe structure based on durability and damage-tolerant design concept Ji-kui Zhang *, Xiao-quan Cheng, Zheng-neng Li School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 5 January 2010 Accepted 29 March 2010 Available online 1 April 2010 Keywords: Total fatigue life Damage-tolerant Titanium alloys
a b s t r a c t Based on the concept of the damage-tolerance and durability design, the total fatigue life of titanium alloys is divided into three phases: crack initiation (0–0.3 mm), short crack growth (0.3–2 mm) and long crack growth (2 mm–aC). Among these three phases, different prediction models are accepted due to different failure mechanisms. A computer program was developed to predict the total fatigue life of the titanium alloy structure. Fatigue testing is also conducted for two types of ELI grade titanium alloy to verify the prediction models. The predicted fatigue life agrees well with experimental results. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Titanium alloys are widely used in the airframe and critical systems for its high specific strength, outstanding resistance to fatigue, high temperature and environmental effects. For example, F-22 air superiority fighter utilizes more than 9000 lb (42 wt%) of titanium alloys. And the extra-low interstitial (ELI) grade of the Ti–6Al–4V (Ti–6–4) is the choice of most safety of flight structure, which has preferable fracture properties over other grade [1]. In other words, design of these structures is based upon the damage-tolerant concept. Fatigue fracture is the main failure mode for these structures. Predictions of total fatigue life are important for safety and economy of these structures. There are two methods for estimation of fatigue life of metallic components. The older and still commonly used method is the S–N curve, even though the drawbacks are well known. In this approach, the safe-life of components are predicted on the basis of experimental results using simple specimens [2]. A number of factors, such as: geometry of component, design of joint, process and load sequence affect the accuracy of prediction. As a result, the accuracy of S–N curve method is relative poor and scatter factor of the safe-life is generally from 4 to 6. Other method to estimate fatigue life is the damage-tolerant crack growth model using linear elastic fracture mechanics (LEFM) theory. This method has high accuracy for the prediction of stable crack growth life [3]. However, crack initial life and short crack growth life cannot be modeled by this method. Some researchers [4–7] tried to overcome this problem by correlating long crack
* Corresponding author. Tel.: +86 10 82338215. E-mail address: [email protected] (J.-k. Zhang). 0261-3069/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2010.03.052
growth data as a function of the effective-stress-intensity-factor (DKeff) and modifying DKeff in the near-threshold regime to fit measured short crack growth rate behavior. For example, Newman et al. [8,9] and Wu et al. [10] developed a plasticity-induced crackclosure model to estimate the total fatigue life of metallic material from assumed initial defects or cracks (10–100 lm) to the final fracture. However, the stress intensity factor with high accuracy is required especially during the short crack growth stage. In recent years, the development of high damage-tolerant titanium alloys, short and long crack growth life share a major portion of the total fatigue life. The extra-low interstitial grade titanium alloys have been widely used in heavily loaded primary structures which are designed on the basis of durability and damage-tolerant concept. In this paper, a new division method and its numerical model are presented to predict the total fatigue life of titanium alloy structure. 2. Partition of total fatigue life Corresponding to the existing prediction model, there are two partition methods for total fatigue life of airframe structure. According to conventional safe-life and damage-tolerant concept, the fatigue failure process can be divided into crack initiation and crack growth stages dependent upon whether there is detectable macro-crack. The other partition method is presented by Newman et al. [9] and Wu et al. [10]. The crack initial life is neglected and total fatigue life may be predicted from assumed initial microcracks to the final fracture. For airframe primary structures made of titanium alloys, the defects of above two partition methods are obvious. In the former one, short crack growth life is merged into crack initial life and
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estimated by S–N curve. However, titanium alloys exhibit ‘‘short crack effect”, and short crack growth life occupies about 20–30% of total fatigue life. As a result, the accuracy of prediction is reduced. In the latter one, the crack growth life is assumed to cover the major part of the fatigue life and the crack initiation period becomes relatively short. However, for the aircraft structures built up from sheet and spar, the crack initiation life may be longer and should be considered as a part of fatigue life until some detectable short cracks are present. The experimental research [11] shows that crack (including short and long crack) growth life occupies almost half of the total fatigue life for damage-tolerant titanium alloy structures. To reduce structure weight further, the short and long crack growth life should be utilized when the titanium alloy structures are designed. Based on the structure durability and damage-tolerant design concept, the total fatigue life is divided into three phases: crack initiation (0–0.3 mm), short crack growth (0.3–2 mm) and long crack growth (2 mm–aC). Above partition is illustrated in Fig. 1. aC is the critical crack length before crack rapidly grows.
2.1. Partition of crack initiation and growth End of crack initiation life is difficult to determine due to its dependence upon the accuracy of inspection instruments, concept of design, the precision of model, etc. It is possible to inspect several micron cracks by thin film replication technique, but this approach is costly and time consuming. Therefore, it is usually applied in laboratory environment only. In this paper, the initial crack size is assumed to be 0.3 mm due to the following reasons. Firstly, the long distance microscope QM-100 system (as shown in Fig. 2) was applied to observe and measure the crack growth behavior in this study. There is perfect detection reliability and measurement accuracy when crack size is over 0.3 mm. Since the system is far from the object to be detected, this could be conveniently applied for airframe structures inspection and maintenance. Secondly, the fractograph analysis results for several hundreds specimen show that the crack growth behavior could be deduced when crack size is up to 0.3 mm. Therefore, 0.3 mm is the partition indication of crack initial phase and short crack growth phase. By integrating the partition method and fractograph estimation technology, it is possible to estimate the occurrence time of ‘‘the detectable short crack in engineering”. It is instructive and useful to make the decision for structure maintenance and repair schedule. Finally, the maximum tolerant initial crack size emanating from a hole is 0.3 mm in durability analysis. Therefore, the fatigue life
crack length/mm
aC
Phase
Phase
Phase
partition method and prediction model could also be applied for structure durability design. 2.2. Partition of short crack growth and long crack growth In general, damage-tolerant structure design concepts focus on the long crack (a > 2 mm) growth life only. So the titanium alloys are regarded as classic high strength and low fracture toughness materials. And it is not suitable to use damage-tolerant design concept. However, the short crack growth behavior of titanium alloys exhibits obviously ‘‘short crack effect”, which is different from that of long crack growth. While suffering the same nominal DK (amplitude of stress intensity factor), short crack grows rapidly than long crack because the ‘‘crack closure effect” is weakened in the short crack. Hence, the life of short and long crack growth should be estimated by different models. In this paper, the crack growth process is divided into two phases: short crack growth and long crack growth. The partition indication for the crack length between the short and long crack growths is defined as 2 mm. The period over which the crack grows from 2 mm to ac denotes the long crack growth life that is highly concerned by structure designer. The period over which the crack grows from 0.3 mm to 2 mm denotes the short crack growth life. The estimation of short crack growth life is significant to make the structural reliable. If a structure is designed on the basis of durability concept, the partition indication of the short and long cracks may also be defined as 1 mm. 3. Total fatigue life prediction model As described above, the total fatigue life of the titanium alloy structures is divided into three different successive periods. The sequence of the periods is: (1) crack initiation (0–0.3 mm); (2) short crack growth (0.3–2 mm); (3) long crack growth (2 mm–ac). Different prediction models have to be adopted for the three periods due to different failure mechanisms. 3.1. Fatigue crack initiation life prediction As mentioned above, the fatigue crack initiation life is defined as the period until the crack length is longer than 0.3 mm. Precise prediction of crack initiation life is very difficult. The mechanism of crack initiation is so complex that the fatigue life can not be estimated by linear elastic or plastic fracture mechanics. For example, the crack initiation size and location are affected by microwave, grain structure and size, specimen geometry, etc. Miner’s linear accumulative damage model is widely used for fatigue life prediction in engineering for its simplicity and availability. Therefore, the crack initiation life could be predicted by means of Miner’s rule [12]:
D¼
k X
nj =Nj ¼ n1 =N1 þ n2 =N2 þ þ nk =N k ¼ 1
ð1Þ
j¼1
2.00 0.30
cycle number Fig. 1. Partition of the total fatigue life Phase I: crack initiation; Phase II: short crack growth; Phase III: long crack growth.
Eq. (1) is the formula from the failure criterion of Miner’s rule. In the formula, nj and Nj represent the actual cyclic number and the constant amplitude cyclic number during crack initiation at a specific amplitude level j, respectively, and k is the total number of discrete amplitude levels. The total damage ratio D is summated by individual relative damage ratio nj/Nj. The initial crack 0.3 mm in length occurs as soon as D P 1. The S–Ni curves, plotting the cycle number Ni during crack initiation against the stress amplitude S, are necessary for predicting
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Fig. 2. QM-100 long distance microscope system for crack observation and measurement (a) QM-100 long distance microscope (b) and (c) short crack growth photographs taken from QM-100.
the fatigue crack initiation life. In general, Ni is obtained by testing. Ni may also be obtained by subtracting the prediction of crack growth life from total fatigue life.
Concerning the prediction of crack growth under variable amplitude loading, the well known modified Willenberg [15] model (Eq. (4)) may be adopted, considering the retardation effect of crack growth.
3.2. Fatigue long crack growth life prediction
8 q p > < Cð1 Reff Þ ðK maxeff Þ da=dN ¼ CðK maxeff Þp > : Cð1 Reff Þr ðK maxeff Þp
Prediction of crack growth life is same for short and long cracks. The only difference is the effective-stress-intensity-factor instead of the stress intensity factor. So the model for long crack growth life prediction is presented firstly. If the crack length is more than 2 mm, long crack growth period starts. Fatigue crack growth primarily depends upon the crack tip driving stress history and crack growth resistance of material. During this period, ‘‘linear elastic fracture mechanics” can be applied, and the crack tip stress intensity factor K becomes an important parameter. The crack tip stress intensity factor is a function of bearing stress, specimen geometry and crack shape et al. During each cycle the crack extension Da is equal to the crack growth rate in that cycle which can be obtained from a calibration curve da/ dN = f(DK, R). The most popular formulae for predicting the crack growth rate under constant amplitude loading are presented by Paris [13] (Eq. (2)) and Walker [14] (Eq. (3)) as:
da=dN ¼ CðDKÞn
ð2Þ
a, n, da/dN are the crack length, number of cycles and crack growth rate, respectively. C and n are the material dependent constants obtained through testing.
9 da=dN ¼ Cfð1 RÞM1 DKgn ¼ Cfð1 RÞM K max gn > > > = 1 > R > 0 M ¼ M1 > 1 < R 6 0 M ¼ M2 > > ; R 6 1 M ¼ 1:1
ð3Þ
R is stress ratio, C, M, M1, M2 and n are the material constants which can be obtained in terms of the curve fitting from experimental data.
Reff > 0 Reff ¼ 0
ð4Þ
Reff < 0
Reff is defined as the effective stress ratio and Kmaxeff denotes the maximum effective-stress-intensity-factor involving the crack closure effect induced by plasticity. C, q, p and r represent the material parameters that can be obtained from curve fitting of experimental data.Suitable model must be selected according to martial properties, structural and loading characteristics, et al. Long crack growth life is given by integrating the selected model. 3.3. Fatigue short crack growth life prediction The short crack growth behavior is more complex as compared to long crack growth. Above threshold value the short crack grows much faster than long crack while having the same nominal DK. As a consequence, the short crack growth life predicted from the experimental data during long crack growth becomes relatively dangerous. Some researchers [16] presented a series of elastic– plastic fracture mechanics concepts and approaches, such as the J-integrity and crack opening displacement theory, to investigate and explain the short crack effects observed. However, the complexity of elastic–plastic fracture mechanics has restricted its application in engineering. In this paper, the plasticity-induced crack-closure model developed by Newman and Phillips [9] is adopted to predict the short crack growth life for metallic materials. While dealing with the short crack growth problem, more attention should be paid to the following two aspects.
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pffiffiffiffiffiffiffiffiffiffiffiffi DK ¼ Dr pa=Q F jn
Crack initiation phase
Firstly, obtaining highly precise solution of stress intensity factor is the most important precondition for prediction of the short crack growth behavior. Most short cracks are non-through surface-crack or corner-crack, and the stress intensity factor could be calculated by correlating Newman–Raju finite element analysis with Zhao-Wu three-dimensional weight function method. The stress intensity factor is given in the following equation.
ð5Þ
Input initial crack a0 and S
Ni curve
Define initial cumulated damage D, cycle number N N=N+n 1
Calculate ni/ Ni due to ni repetitions of σ i
Dr is the range of variation of stress, Q and Fjn are the modification factors to consider shape and boundary condition effects, respectively. The detail calculation of Q, and Fjn could be found in Ref. [10]. Secondly, the closure effect of a short crack is not similar to that of long crack. Crack growth rate of a short crack will be greater than steady-state crack growth rates. Considering the crack closure effect, an amplitude of effective-stress-intensity-factor DKeff is introduced to predict the short crack growth behavior and given by
N=N+ni
DK eff ¼ DKð1 rop =rmax Þ=ð1 RÞ
Calculate Δ ~K d eff d~ d a d N curves by Eq.(6)
rmax is the maximum stress value, R is stress ratio, and rop is the crack opening stress and given by
(
rop =rmax ¼ A0 þ A1 R þ A2 R2 þ A3 R3 R P 0 rop =rmax ¼ A0 þ A1 R R<0
ð7Þ
In which, the calculation of A0, A1, A2, and A3 could be obtained from Ref. [17] in details.
N
D≥ 1 Y a=0.3mm
Short crack growth phase
ð6Þ
Calculate cumulated damage by Eq.(1)
N=N+ni Calculated procedure same to long crack N
a>2mm Y
3.4. Total fatigue life prediction model
4. Experimental research To verify the total fatigue life prediction model presented in this paper, fatigue testing was conducted for two types of ELI grade titanium alloys: TC4ELI and TA15ELI. These have the typical coarse lamellar microstructures, as shown in Fig. 4. Both of the crack growth trajectories are inflexed, which decreases the crack growth rate and increases the fracture toughness. As a result, TC4ELI and TA15ELI have better damage-tolerant properties than common titanium alloys. Fatigue testing was conducted on Instron 8802 servo-hydraulic test systems at room temperature. The specimen configuration is shown in Fig. 5. All the specimens have with a 4 mm diameter hole at centre. Hole edge is filleted on one side (as shown in Fig. 5). The cracks initiate on the other side and may be observed by QM-100 long distance microscope system, as shown in Fig. 2. The testing results for the specimens under constant amplitude stress rmax = 320 MPa, R = 0.1 are given in Fig. 6. The results for the specimens under variable amplitude stress are shown in Fig. 7. It should be noted that the fatigue life in Figs. 6 and 7 is the average value of five pieces of specimens testing results. As shown in Figs. 6 and 7, crack initial life occupies the most portion of the total fatigue life: 70% under constant amplitude stress; 60% under variable amplitude stress. Long crack growth life occupies the least portion, 10–15% for either constant or variable amplitude stress. Short crack growth life occupies a little more por-
Long crack growth phase
To predict the total fatigue life of metallic materials, a computer program has been developed using Fortran code. The procedure of the program is explained by the flow chart, as shown in Fig. 3.
Input material properties of long crack growth N=N+n Yi CalculateΔK Calculate Δa by one of Eq(2),(3),(4) a=a+Δa N
> acc a>a a Y End
Fig. 3. Flow chart of the total fatigue life prediction model.
tion than that of long crack, 15–20% for constant amplitude stress and 20–25% for variable amplitude stress. 5. Prediction of total fatigue life Using the program developed, total fatigue life of TC4ELI and TA15ELI is predicted and compared with experimental results. Material properties are given in Table 1. The crack growth life under constant amplitude stress was estimated by Walker’s formula. Crack growth life under variable amplitude stress was estimated by modified Willonberg’s model.
J.-k. Zhang et al. / Materials and Design 31 (2010) 4329–4335
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Fig. 4. Microstructure of TC4ELI and TA15ELI.
Fig. 5. Specimen configuration.
5.1. Prediction of the fatigue life under constant amplitude loading
life/cycle
8.E+04 7.E+04
TC4ELI
6.E+04
TA15ELI
The comparison between predicted results and testing data for total fatigue life under constant amplitude loading is shown in Table 2. From this table, the following conclusions were obtained.
5.E+04 4.E+04 3.E+04 2.E+04 1.E+04 0.E+00 initial
short
long
total
crack growth phase Fig. 6. Results for specimen with constant amplitude stress.
7.E+05 6.E+05
TC4ELI TA15ELI
life /cycle
5.E+05
(1) Crack initiation life occupies the most proportion of the total fatigue life and the accuracy of prediction is relatively poor. The basic reason is that the fatigue crack initiation is a highly complex and random process and cannot be characterized by a single damage parameter in Miner’s linear accumulative damage model. However, the relative error of prediction for crack initiation life is within 25%, which still meets the engineering requirement. (2) Long crack propagation life occupies the least proportion of total fatigue life and the error of prediction is less. The DK obtained from linear elastic fracture mechanics has high accuracy during long crack growth phase. (3) Short crack growth life is greater than that of long crack growth life, but accuracy of prediction is less than long one. The plasticity-induced crack-closure model might be required to estimate the short crack growth behavior.
4.E+05 3.E+05
5.2. Prediction of the fatigue life under variable amplitude loading
2.E+05 1.E+05 0.E+00 initial
short
long
total
crack growth phase Fig. 7. Results for specimen with variable amplitude stress.
Table 3 gives the comparison between predicted results and test results for total fatigue life under variable amplitude loading. Crack initial life has the lowest accuracy: prediction error over 30%. Prediction error for short crack growth life exceeds 20% also. For long crack growth life, the prediction error is within 20%. As a whole, the total fatigue life could be estimated with the prediction error less than 25%.
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Table 1 Basic mechanical properties of TC4ELI and TA15ELI.
rb (MPa)
Material TC4ELI TA15ELI
858 980
KIC (MPaM0.5)
r0.2 (MPa) 794 900
Materials
Fatigue phase
Life Tested/ cycle
Predicted/ cycle
Relative error (%)
TC4ELI
Initial Short Long Total
50,167 11,667 9433 71,267
38,500 9759 9172 57,431
23.26 16.35 2.77 19.41
TA15ELI
Initial Short Long Total
54,333 14,000 10,000 78,333
61,500 15,311 10,320 87,131
13.19 9.36 3.20 11.23
Table 3 Predicted and tested fatigue lives of TC4ELI and TA15ELI specimen under spectrum loading. Materials
8
Fatigue phase
Life Tested/ cycle
Predicted/ cycle
Relative error (%)
TC4ELI
Initial Short Long Total
224,746 98,190 56,552 379,488
154,459 76,140 67,100 297,699
31.27 22.46 18.65 21.55
TA15ELI
Initial Short Long Total
386,214 140,098 77,002 603,314
263,600 108,000 81,300 452,900
31.75 22.91 5.58 24.93
Comparison of Tables 2 and 3 shows that, the prediction accuracy of the specimen under variable amplitude loading is less than those under constant amplitude loading for every phase of fatigue life. The difference may be due to the interaction among high and low loading levels. 6. Discussion 6.1. Total fatigue life partition method On the basis of durability and damage-tolerant concept, a model for prediction of the total fatigue life of airframe structure is developed. The total fatigue life of Ti-alloys is divided into three phases: crack initiation, short crack growth and long crack growth. The advantages of the division methods could be listed as below. Firstly, comparison with the conventional damage-tolerant crack division method used in most Ref. [18], the most prominent merit is that the approximate time of ‘‘engineering detectable crack” occurrence may be found. It is useful to make structure inspection schedule and crucial to durability and damage-tolerant design of Ti-alloys airframe structure. Secondly, for each fatigue phase of Ti-alloys airframe structure, the prediction model is established according to respective failure mechanism: Miner’s rule for crack initiation, classic linear elastic fracture mechanics theory for long crack growth and the plasticity-induced crack-closure model for short crack growth. It is more
2.74
da/dN = 5.28 10 (DK) da/dN = 1.76 108 (DK)2.85
106 111
Table 2 Predicted and tested fatigue lives for TC4ELI and TA15ELI specimens under constant amplitude loading.
DKth (MPaM0.5)
Walk formula
4.9 5.1
widespread than traditional linear elastic fracture mechanics theory [3]. Simultaneously, it is more convenient and efficient than those elastic–plastic fracture mechanics approaches [16]. Thirdly, in this paper the initial crack size is assumed to be 0.3 mm rather than several microns in Refs. [8,9]. Therefore, thin film replication technique is free to detect the micro-crack. Correspondingly, the long distance microscope QM-100 system, as shown in Fig. 2, is allowable to detect the short crack. That means the accuracy requirement of the detection instrument in this paper is not so high as that in Ref. [10]. That could reduce the time and cost of the crack detection. Finally, in this paper, 0.3 mm and 2 mm are defined as the partition indications for titanium alloys. However, it should be noted that the partition indication could not be determined exactly. In fact, it is mainly dependent upon the design concept and the detective capability of instruments.
6.2. Total fatigue life prediction model Based on the separation method presented above, total fatigue life prediction model can be established. The prediction results agree quite well with experimental data, as shown in Tables 2 and 3. Although the prediction model is only applied to two types of ELI grade titanium alloys, it could also be used for other types of materials after necessary modification. The modification involves adopting suitable crack detection and measurement instrument, selecting appropriate crack growth rate formula, supplying indispensable material parameter, and so on. In summary, the prediction model has adequate accuracy and practical value for the structure design based on damage-tolerance and durability concept.
7. Conclusion Based on the durability and damage-tolerant design concept, a total fatigue life separation method and corresponding prediction model are presented. The conclusions from our research could be summarized as below. According to the requirement of structural design and the development of crack inspection technology, the total fatigue life is divided into three phases: crack initiation, short crack growth and long crack growth. For titanium alloys, 0.3 mm and 2 mm are defined as the partition indications. Fatigue testing was conducted for two types of titanium alloys. The experimental results show that crack initial life, short crack growth life and long crack growth life occupy about 60%, 25% and 15% of total life, respectively. Comparison of experimental data and prediction results show that a good agreement for total fatigue life prediction for titanium alloys is achieved.
Acknowledgements This project was financially supported by Fan-Zhou foundation (No. 20060502). The testing specimen is provided by Institute of Metal Research, Chinese Academy of Science. Professor Chang-He Kou gave some valuable advices regarding experimentation.
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References [1] James DC, Larry PC, Herry RP. Titanium alloys on the F-22 fighter airframe. Adv Mater Process 2002;160(5):25–8. [2] Makkonen M. Predicting the total fatigue life in metals. Int J Fatigue 2009;31(7):1163–75. [3] Schijve J. Fatigue damage in aircraft structures, not wanted, but tolerated? Int J Fatigue 2009;31(6):998–1011. [4] Vormwald M, Seeger T. The consequences of short crack closure on fatigue crack growth under variable amplitude loading. Fatigue Fract Eng Mater Struct 1991;14(2–3):205–25. [5] Endo M, McEvily AJ. Prediction of the behavior of small fatigue cracks. Mater Sci Eng A 2007;468–470:51–8. [6] Halliday MD, Cooper C, Poole P, et al. On predicting small fatigue crack growth and fatigue life from long crack data in 2024 aluminum alloy. Int J Fatigue 2003;25(8):709–18. [7] Garcia DB, Grandt AF. Application of a total life prediction model for fretting fatigue in Ti–6Al–4V. Int J Fatigue 2007;29(7):1311–8. [8] Newman JC, Ruschau JJ. The stress-level effect on fatigue-crack growth under constant-amplitude loading. Int J Fatigue 2007;29(9–11):1608–15. [9] Newman JC, Phillips EP, Swain MH. Fatigue-life prediction methodology using small-crack theory. Int J Fatigue 1999;21(2):109–19.
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[10] Wu XR, Newman JC, Zhao W, et al. Small crack growth and fatigue life predictions for high-strength aluminum alloys: part I – experimental and fracture mechanics analysis. Fatigue Fract Eng Mater Struct 1998;21(11): 1289–306. [11] Zhang JK, Li ZN, Qiu ZP, et al. Feasibility study on damage tolerance design of titanium alloys. Acta Aeronaut Astronaut Sin 2009;30(4):763–7 [in Chinese]. [12] Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;12(3):159–64. [13] Paris PC, Gomez MP, Anderson WP. A rational analytic theory of fatigue. Trend Eng 1961;13:9–14. [14] Walker EK. The effect of stress ratio during crack propagation, fatigue for 2024 –T3 and 7075–T6 aluminum. Effects of environment and complex load history on fatigue life. ASTM STP 462 1970:1–14. [15] Willenberg J, Engle RM, Wood HA. A crack growth retardation model using an effective stress concept. AFFDL-TM-71-1-FBR: A715659. [16] El Haddad MH, Dowling NE, Topper TH, et al. J integral applications for short fatigue cracks at notches. Int J Fract 1980;16(1):15–30. [17] Ding ChuanFu, Liu JianZhong, Wu XueRen. An investigation of small-crack and long-crack propagation behavior in titanium alloy TC4 and aluminum alloy 7475-T7351. J Aeronaut Mater 2005;25(6):11–7 [in Chinese]. [18] Jan S, Ivan N, Jeri K. Fatigue crack growth history in damage tolerance design of aircraft structures. Int J Fatigue 2009;31(6):1062–7.
Contents lists available at ScienceDirect
Materials and Design journal homepage: www.elsevier.com/locate/matdes
Total fatigue life prediction for Ti-alloys airframe structure based on durability and damage-tolerant design concept Ji-kui Zhang *, Xiao-quan Cheng, Zheng-neng Li School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 5 January 2010 Accepted 29 March 2010 Available online 1 April 2010 Keywords: Total fatigue life Damage-tolerant Titanium alloys
a b s t r a c t Based on the concept of the damage-tolerance and durability design, the total fatigue life of titanium alloys is divided into three phases: crack initiation (0–0.3 mm), short crack growth (0.3–2 mm) and long crack growth (2 mm–aC). Among these three phases, different prediction models are accepted due to different failure mechanisms. A computer program was developed to predict the total fatigue life of the titanium alloy structure. Fatigue testing is also conducted for two types of ELI grade titanium alloy to verify the prediction models. The predicted fatigue life agrees well with experimental results. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Titanium alloys are widely used in the airframe and critical systems for its high specific strength, outstanding resistance to fatigue, high temperature and environmental effects. For example, F-22 air superiority fighter utilizes more than 9000 lb (42 wt%) of titanium alloys. And the extra-low interstitial (ELI) grade of the Ti–6Al–4V (Ti–6–4) is the choice of most safety of flight structure, which has preferable fracture properties over other grade [1]. In other words, design of these structures is based upon the damage-tolerant concept. Fatigue fracture is the main failure mode for these structures. Predictions of total fatigue life are important for safety and economy of these structures. There are two methods for estimation of fatigue life of metallic components. The older and still commonly used method is the S–N curve, even though the drawbacks are well known. In this approach, the safe-life of components are predicted on the basis of experimental results using simple specimens [2]. A number of factors, such as: geometry of component, design of joint, process and load sequence affect the accuracy of prediction. As a result, the accuracy of S–N curve method is relative poor and scatter factor of the safe-life is generally from 4 to 6. Other method to estimate fatigue life is the damage-tolerant crack growth model using linear elastic fracture mechanics (LEFM) theory. This method has high accuracy for the prediction of stable crack growth life [3]. However, crack initial life and short crack growth life cannot be modeled by this method. Some researchers [4–7] tried to overcome this problem by correlating long crack
* Corresponding author. Tel.: +86 10 82338215. E-mail address: [email protected] (J.-k. Zhang). 0261-3069/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2010.03.052
growth data as a function of the effective-stress-intensity-factor (DKeff) and modifying DKeff in the near-threshold regime to fit measured short crack growth rate behavior. For example, Newman et al. [8,9] and Wu et al. [10] developed a plasticity-induced crackclosure model to estimate the total fatigue life of metallic material from assumed initial defects or cracks (10–100 lm) to the final fracture. However, the stress intensity factor with high accuracy is required especially during the short crack growth stage. In recent years, the development of high damage-tolerant titanium alloys, short and long crack growth life share a major portion of the total fatigue life. The extra-low interstitial grade titanium alloys have been widely used in heavily loaded primary structures which are designed on the basis of durability and damage-tolerant concept. In this paper, a new division method and its numerical model are presented to predict the total fatigue life of titanium alloy structure. 2. Partition of total fatigue life Corresponding to the existing prediction model, there are two partition methods for total fatigue life of airframe structure. According to conventional safe-life and damage-tolerant concept, the fatigue failure process can be divided into crack initiation and crack growth stages dependent upon whether there is detectable macro-crack. The other partition method is presented by Newman et al. [9] and Wu et al. [10]. The crack initial life is neglected and total fatigue life may be predicted from assumed initial microcracks to the final fracture. For airframe primary structures made of titanium alloys, the defects of above two partition methods are obvious. In the former one, short crack growth life is merged into crack initial life and
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estimated by S–N curve. However, titanium alloys exhibit ‘‘short crack effect”, and short crack growth life occupies about 20–30% of total fatigue life. As a result, the accuracy of prediction is reduced. In the latter one, the crack growth life is assumed to cover the major part of the fatigue life and the crack initiation period becomes relatively short. However, for the aircraft structures built up from sheet and spar, the crack initiation life may be longer and should be considered as a part of fatigue life until some detectable short cracks are present. The experimental research [11] shows that crack (including short and long crack) growth life occupies almost half of the total fatigue life for damage-tolerant titanium alloy structures. To reduce structure weight further, the short and long crack growth life should be utilized when the titanium alloy structures are designed. Based on the structure durability and damage-tolerant design concept, the total fatigue life is divided into three phases: crack initiation (0–0.3 mm), short crack growth (0.3–2 mm) and long crack growth (2 mm–aC). Above partition is illustrated in Fig. 1. aC is the critical crack length before crack rapidly grows.
2.1. Partition of crack initiation and growth End of crack initiation life is difficult to determine due to its dependence upon the accuracy of inspection instruments, concept of design, the precision of model, etc. It is possible to inspect several micron cracks by thin film replication technique, but this approach is costly and time consuming. Therefore, it is usually applied in laboratory environment only. In this paper, the initial crack size is assumed to be 0.3 mm due to the following reasons. Firstly, the long distance microscope QM-100 system (as shown in Fig. 2) was applied to observe and measure the crack growth behavior in this study. There is perfect detection reliability and measurement accuracy when crack size is over 0.3 mm. Since the system is far from the object to be detected, this could be conveniently applied for airframe structures inspection and maintenance. Secondly, the fractograph analysis results for several hundreds specimen show that the crack growth behavior could be deduced when crack size is up to 0.3 mm. Therefore, 0.3 mm is the partition indication of crack initial phase and short crack growth phase. By integrating the partition method and fractograph estimation technology, it is possible to estimate the occurrence time of ‘‘the detectable short crack in engineering”. It is instructive and useful to make the decision for structure maintenance and repair schedule. Finally, the maximum tolerant initial crack size emanating from a hole is 0.3 mm in durability analysis. Therefore, the fatigue life
crack length/mm
aC
Phase
Phase
Phase
partition method and prediction model could also be applied for structure durability design. 2.2. Partition of short crack growth and long crack growth In general, damage-tolerant structure design concepts focus on the long crack (a > 2 mm) growth life only. So the titanium alloys are regarded as classic high strength and low fracture toughness materials. And it is not suitable to use damage-tolerant design concept. However, the short crack growth behavior of titanium alloys exhibits obviously ‘‘short crack effect”, which is different from that of long crack growth. While suffering the same nominal DK (amplitude of stress intensity factor), short crack grows rapidly than long crack because the ‘‘crack closure effect” is weakened in the short crack. Hence, the life of short and long crack growth should be estimated by different models. In this paper, the crack growth process is divided into two phases: short crack growth and long crack growth. The partition indication for the crack length between the short and long crack growths is defined as 2 mm. The period over which the crack grows from 2 mm to ac denotes the long crack growth life that is highly concerned by structure designer. The period over which the crack grows from 0.3 mm to 2 mm denotes the short crack growth life. The estimation of short crack growth life is significant to make the structural reliable. If a structure is designed on the basis of durability concept, the partition indication of the short and long cracks may also be defined as 1 mm. 3. Total fatigue life prediction model As described above, the total fatigue life of the titanium alloy structures is divided into three different successive periods. The sequence of the periods is: (1) crack initiation (0–0.3 mm); (2) short crack growth (0.3–2 mm); (3) long crack growth (2 mm–ac). Different prediction models have to be adopted for the three periods due to different failure mechanisms. 3.1. Fatigue crack initiation life prediction As mentioned above, the fatigue crack initiation life is defined as the period until the crack length is longer than 0.3 mm. Precise prediction of crack initiation life is very difficult. The mechanism of crack initiation is so complex that the fatigue life can not be estimated by linear elastic or plastic fracture mechanics. For example, the crack initiation size and location are affected by microwave, grain structure and size, specimen geometry, etc. Miner’s linear accumulative damage model is widely used for fatigue life prediction in engineering for its simplicity and availability. Therefore, the crack initiation life could be predicted by means of Miner’s rule [12]:
D¼
k X
nj =Nj ¼ n1 =N1 þ n2 =N2 þ þ nk =N k ¼ 1
ð1Þ
j¼1
2.00 0.30
cycle number Fig. 1. Partition of the total fatigue life Phase I: crack initiation; Phase II: short crack growth; Phase III: long crack growth.
Eq. (1) is the formula from the failure criterion of Miner’s rule. In the formula, nj and Nj represent the actual cyclic number and the constant amplitude cyclic number during crack initiation at a specific amplitude level j, respectively, and k is the total number of discrete amplitude levels. The total damage ratio D is summated by individual relative damage ratio nj/Nj. The initial crack 0.3 mm in length occurs as soon as D P 1. The S–Ni curves, plotting the cycle number Ni during crack initiation against the stress amplitude S, are necessary for predicting
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Fig. 2. QM-100 long distance microscope system for crack observation and measurement (a) QM-100 long distance microscope (b) and (c) short crack growth photographs taken from QM-100.
the fatigue crack initiation life. In general, Ni is obtained by testing. Ni may also be obtained by subtracting the prediction of crack growth life from total fatigue life.
Concerning the prediction of crack growth under variable amplitude loading, the well known modified Willenberg [15] model (Eq. (4)) may be adopted, considering the retardation effect of crack growth.
3.2. Fatigue long crack growth life prediction
8 q p > < Cð1 Reff Þ ðK maxeff Þ da=dN ¼ CðK maxeff Þp > : Cð1 Reff Þr ðK maxeff Þp
Prediction of crack growth life is same for short and long cracks. The only difference is the effective-stress-intensity-factor instead of the stress intensity factor. So the model for long crack growth life prediction is presented firstly. If the crack length is more than 2 mm, long crack growth period starts. Fatigue crack growth primarily depends upon the crack tip driving stress history and crack growth resistance of material. During this period, ‘‘linear elastic fracture mechanics” can be applied, and the crack tip stress intensity factor K becomes an important parameter. The crack tip stress intensity factor is a function of bearing stress, specimen geometry and crack shape et al. During each cycle the crack extension Da is equal to the crack growth rate in that cycle which can be obtained from a calibration curve da/ dN = f(DK, R). The most popular formulae for predicting the crack growth rate under constant amplitude loading are presented by Paris [13] (Eq. (2)) and Walker [14] (Eq. (3)) as:
da=dN ¼ CðDKÞn
ð2Þ
a, n, da/dN are the crack length, number of cycles and crack growth rate, respectively. C and n are the material dependent constants obtained through testing.
9 da=dN ¼ Cfð1 RÞM1 DKgn ¼ Cfð1 RÞM K max gn > > > = 1 > R > 0 M ¼ M1 > 1 < R 6 0 M ¼ M2 > > ; R 6 1 M ¼ 1:1
ð3Þ
R is stress ratio, C, M, M1, M2 and n are the material constants which can be obtained in terms of the curve fitting from experimental data.
Reff > 0 Reff ¼ 0
ð4Þ
Reff < 0
Reff is defined as the effective stress ratio and Kmaxeff denotes the maximum effective-stress-intensity-factor involving the crack closure effect induced by plasticity. C, q, p and r represent the material parameters that can be obtained from curve fitting of experimental data.Suitable model must be selected according to martial properties, structural and loading characteristics, et al. Long crack growth life is given by integrating the selected model. 3.3. Fatigue short crack growth life prediction The short crack growth behavior is more complex as compared to long crack growth. Above threshold value the short crack grows much faster than long crack while having the same nominal DK. As a consequence, the short crack growth life predicted from the experimental data during long crack growth becomes relatively dangerous. Some researchers [16] presented a series of elastic– plastic fracture mechanics concepts and approaches, such as the J-integrity and crack opening displacement theory, to investigate and explain the short crack effects observed. However, the complexity of elastic–plastic fracture mechanics has restricted its application in engineering. In this paper, the plasticity-induced crack-closure model developed by Newman and Phillips [9] is adopted to predict the short crack growth life for metallic materials. While dealing with the short crack growth problem, more attention should be paid to the following two aspects.
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pffiffiffiffiffiffiffiffiffiffiffiffi DK ¼ Dr pa=Q F jn
Crack initiation phase
Firstly, obtaining highly precise solution of stress intensity factor is the most important precondition for prediction of the short crack growth behavior. Most short cracks are non-through surface-crack or corner-crack, and the stress intensity factor could be calculated by correlating Newman–Raju finite element analysis with Zhao-Wu three-dimensional weight function method. The stress intensity factor is given in the following equation.
ð5Þ
Input initial crack a0 and S
Ni curve
Define initial cumulated damage D, cycle number N N=N+n 1
Calculate ni/ Ni due to ni repetitions of σ i
Dr is the range of variation of stress, Q and Fjn are the modification factors to consider shape and boundary condition effects, respectively. The detail calculation of Q, and Fjn could be found in Ref. [10]. Secondly, the closure effect of a short crack is not similar to that of long crack. Crack growth rate of a short crack will be greater than steady-state crack growth rates. Considering the crack closure effect, an amplitude of effective-stress-intensity-factor DKeff is introduced to predict the short crack growth behavior and given by
N=N+ni
DK eff ¼ DKð1 rop =rmax Þ=ð1 RÞ
Calculate Δ ~K d eff d~ d a d N curves by Eq.(6)
rmax is the maximum stress value, R is stress ratio, and rop is the crack opening stress and given by
(
rop =rmax ¼ A0 þ A1 R þ A2 R2 þ A3 R3 R P 0 rop =rmax ¼ A0 þ A1 R R<0
ð7Þ
In which, the calculation of A0, A1, A2, and A3 could be obtained from Ref. [17] in details.
N
D≥ 1 Y a=0.3mm
Short crack growth phase
ð6Þ
Calculate cumulated damage by Eq.(1)
N=N+ni Calculated procedure same to long crack N
a>2mm Y
3.4. Total fatigue life prediction model
4. Experimental research To verify the total fatigue life prediction model presented in this paper, fatigue testing was conducted for two types of ELI grade titanium alloys: TC4ELI and TA15ELI. These have the typical coarse lamellar microstructures, as shown in Fig. 4. Both of the crack growth trajectories are inflexed, which decreases the crack growth rate and increases the fracture toughness. As a result, TC4ELI and TA15ELI have better damage-tolerant properties than common titanium alloys. Fatigue testing was conducted on Instron 8802 servo-hydraulic test systems at room temperature. The specimen configuration is shown in Fig. 5. All the specimens have with a 4 mm diameter hole at centre. Hole edge is filleted on one side (as shown in Fig. 5). The cracks initiate on the other side and may be observed by QM-100 long distance microscope system, as shown in Fig. 2. The testing results for the specimens under constant amplitude stress rmax = 320 MPa, R = 0.1 are given in Fig. 6. The results for the specimens under variable amplitude stress are shown in Fig. 7. It should be noted that the fatigue life in Figs. 6 and 7 is the average value of five pieces of specimens testing results. As shown in Figs. 6 and 7, crack initial life occupies the most portion of the total fatigue life: 70% under constant amplitude stress; 60% under variable amplitude stress. Long crack growth life occupies the least portion, 10–15% for either constant or variable amplitude stress. Short crack growth life occupies a little more por-
Long crack growth phase
To predict the total fatigue life of metallic materials, a computer program has been developed using Fortran code. The procedure of the program is explained by the flow chart, as shown in Fig. 3.
Input material properties of long crack growth N=N+n Yi CalculateΔK Calculate Δa by one of Eq(2),(3),(4) a=a+Δa N
> acc a>a a Y End
Fig. 3. Flow chart of the total fatigue life prediction model.
tion than that of long crack, 15–20% for constant amplitude stress and 20–25% for variable amplitude stress. 5. Prediction of total fatigue life Using the program developed, total fatigue life of TC4ELI and TA15ELI is predicted and compared with experimental results. Material properties are given in Table 1. The crack growth life under constant amplitude stress was estimated by Walker’s formula. Crack growth life under variable amplitude stress was estimated by modified Willonberg’s model.
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Fig. 4. Microstructure of TC4ELI and TA15ELI.
Fig. 5. Specimen configuration.
5.1. Prediction of the fatigue life under constant amplitude loading
life/cycle
8.E+04 7.E+04
TC4ELI
6.E+04
TA15ELI
The comparison between predicted results and testing data for total fatigue life under constant amplitude loading is shown in Table 2. From this table, the following conclusions were obtained.
5.E+04 4.E+04 3.E+04 2.E+04 1.E+04 0.E+00 initial
short
long
total
crack growth phase Fig. 6. Results for specimen with constant amplitude stress.
7.E+05 6.E+05
TC4ELI TA15ELI
life /cycle
5.E+05
(1) Crack initiation life occupies the most proportion of the total fatigue life and the accuracy of prediction is relatively poor. The basic reason is that the fatigue crack initiation is a highly complex and random process and cannot be characterized by a single damage parameter in Miner’s linear accumulative damage model. However, the relative error of prediction for crack initiation life is within 25%, which still meets the engineering requirement. (2) Long crack propagation life occupies the least proportion of total fatigue life and the error of prediction is less. The DK obtained from linear elastic fracture mechanics has high accuracy during long crack growth phase. (3) Short crack growth life is greater than that of long crack growth life, but accuracy of prediction is less than long one. The plasticity-induced crack-closure model might be required to estimate the short crack growth behavior.
4.E+05 3.E+05
5.2. Prediction of the fatigue life under variable amplitude loading
2.E+05 1.E+05 0.E+00 initial
short
long
total
crack growth phase Fig. 7. Results for specimen with variable amplitude stress.
Table 3 gives the comparison between predicted results and test results for total fatigue life under variable amplitude loading. Crack initial life has the lowest accuracy: prediction error over 30%. Prediction error for short crack growth life exceeds 20% also. For long crack growth life, the prediction error is within 20%. As a whole, the total fatigue life could be estimated with the prediction error less than 25%.
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Table 1 Basic mechanical properties of TC4ELI and TA15ELI.
rb (MPa)
Material TC4ELI TA15ELI
858 980
KIC (MPaM0.5)
r0.2 (MPa) 794 900
Materials
Fatigue phase
Life Tested/ cycle
Predicted/ cycle
Relative error (%)
TC4ELI
Initial Short Long Total
50,167 11,667 9433 71,267
38,500 9759 9172 57,431
23.26 16.35 2.77 19.41
TA15ELI
Initial Short Long Total
54,333 14,000 10,000 78,333
61,500 15,311 10,320 87,131
13.19 9.36 3.20 11.23
Table 3 Predicted and tested fatigue lives of TC4ELI and TA15ELI specimen under spectrum loading. Materials
8
Fatigue phase
Life Tested/ cycle
Predicted/ cycle
Relative error (%)
TC4ELI
Initial Short Long Total
224,746 98,190 56,552 379,488
154,459 76,140 67,100 297,699
31.27 22.46 18.65 21.55
TA15ELI
Initial Short Long Total
386,214 140,098 77,002 603,314
263,600 108,000 81,300 452,900
31.75 22.91 5.58 24.93
Comparison of Tables 2 and 3 shows that, the prediction accuracy of the specimen under variable amplitude loading is less than those under constant amplitude loading for every phase of fatigue life. The difference may be due to the interaction among high and low loading levels. 6. Discussion 6.1. Total fatigue life partition method On the basis of durability and damage-tolerant concept, a model for prediction of the total fatigue life of airframe structure is developed. The total fatigue life of Ti-alloys is divided into three phases: crack initiation, short crack growth and long crack growth. The advantages of the division methods could be listed as below. Firstly, comparison with the conventional damage-tolerant crack division method used in most Ref. [18], the most prominent merit is that the approximate time of ‘‘engineering detectable crack” occurrence may be found. It is useful to make structure inspection schedule and crucial to durability and damage-tolerant design of Ti-alloys airframe structure. Secondly, for each fatigue phase of Ti-alloys airframe structure, the prediction model is established according to respective failure mechanism: Miner’s rule for crack initiation, classic linear elastic fracture mechanics theory for long crack growth and the plasticity-induced crack-closure model for short crack growth. It is more
2.74
da/dN = 5.28 10 (DK) da/dN = 1.76 108 (DK)2.85
106 111
Table 2 Predicted and tested fatigue lives for TC4ELI and TA15ELI specimens under constant amplitude loading.
DKth (MPaM0.5)
Walk formula
4.9 5.1
widespread than traditional linear elastic fracture mechanics theory [3]. Simultaneously, it is more convenient and efficient than those elastic–plastic fracture mechanics approaches [16]. Thirdly, in this paper the initial crack size is assumed to be 0.3 mm rather than several microns in Refs. [8,9]. Therefore, thin film replication technique is free to detect the micro-crack. Correspondingly, the long distance microscope QM-100 system, as shown in Fig. 2, is allowable to detect the short crack. That means the accuracy requirement of the detection instrument in this paper is not so high as that in Ref. [10]. That could reduce the time and cost of the crack detection. Finally, in this paper, 0.3 mm and 2 mm are defined as the partition indications for titanium alloys. However, it should be noted that the partition indication could not be determined exactly. In fact, it is mainly dependent upon the design concept and the detective capability of instruments.
6.2. Total fatigue life prediction model Based on the separation method presented above, total fatigue life prediction model can be established. The prediction results agree quite well with experimental data, as shown in Tables 2 and 3. Although the prediction model is only applied to two types of ELI grade titanium alloys, it could also be used for other types of materials after necessary modification. The modification involves adopting suitable crack detection and measurement instrument, selecting appropriate crack growth rate formula, supplying indispensable material parameter, and so on. In summary, the prediction model has adequate accuracy and practical value for the structure design based on damage-tolerance and durability concept.
7. Conclusion Based on the durability and damage-tolerant design concept, a total fatigue life separation method and corresponding prediction model are presented. The conclusions from our research could be summarized as below. According to the requirement of structural design and the development of crack inspection technology, the total fatigue life is divided into three phases: crack initiation, short crack growth and long crack growth. For titanium alloys, 0.3 mm and 2 mm are defined as the partition indications. Fatigue testing was conducted for two types of titanium alloys. The experimental results show that crack initial life, short crack growth life and long crack growth life occupy about 60%, 25% and 15% of total life, respectively. Comparison of experimental data and prediction results show that a good agreement for total fatigue life prediction for titanium alloys is achieved.
Acknowledgements This project was financially supported by Fan-Zhou foundation (No. 20060502). The testing specimen is provided by Institute of Metal Research, Chinese Academy of Science. Professor Chang-He Kou gave some valuable advices regarding experimentation.
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