Prediction of semi-elliptical surface crack growth in 2024-T4 aluminium alloy

International Journal of Pressure Vessels and Piping 79 (2002) 273±278

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Prediction of semi-elliptical surface crack growth in 2024-T4 aluminium alloy P.S. Song a,*, B.C. Sheu b, Y.L. Shieh b a

b

Department of Civil Engineering, Dahan Institute of Technology, Sincheng, Hualien 971, Taiwan, ROC Department of System Engineering, Chung Cheng Institute of Technology, National Defense University, Tahsi, Taoyuan 335 Taiwan, ROC Received 22 November 2001; revised 21 February 2002; accepted 21 February 2002

Abstract This research investigated how a semi-elliptical surface crack grows in a 2024-T4 aluminium alloy under constant-amplitude loading. For the investigation, Sih's energy approach for predicting through-thickness crack growth was generalized to predict surface crack growth. The generalized Sih's energy approach accurately predicted experimental results and the prediction of the Paris equation in the depth direction. Additionally, the generalized approach accurately predicted fatigue life of the surface crack. Finally, a model was established to predict crack shape development by using the measured surface length. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Surface crack; Energy approach; Crack shape

1. Introduction

2. Surface crack growth

A surface crack may escape detection in structural components such as pressure vessels and piping systems. Under cyclic pressurization, the surface crack evolves into a through-thickness crack, increasing the risk of fatigue failure and reducing the useful service life. To reduce failures, surface crack behaviour in structural components has been extensively investigated [1±4]. For such investigations, sizing both the crack depth and the surface length is a prime concern. However, the depth of the crack is dif®cult to measure. To overcome this dif®culty, various nondestructive techniques have been employed, including the alternating current potential drop [5], alternating current ®eld measurement [6], direct current potential drop [7], etc. Apart from these techniques, different crack growth models have been used to predict the crack depth. Among these, the Paris equation is most often used [8±10]. In the ®eld of fatigue, Sih employed an energy approach and closely predicted the growth behaviour of a throughthickness crack [11±13]. However, Sih's work paid little attention to the prediction of surface crack propagation. In this paper, the concept of Sih's approach is extended to predict the development of a surface crack, and the resulting predictions are compared with Paris predictions and experimental results.

2.1. Paris law

* Corresponding author. Tel./fax: 1886-3-8263936. E-mail address: [email protected] (P.S. Song).

For simplicity, the shape of a surface crack is often considered as semi-elliptical, as shown in Fig. 1; where a is the crack depth, b is half the surface length, W is half the plate width, and t is the plate thickness. For the crack, the Paris law gives the growth rates in the crack depth and surface length directions as follows [14] da ˆ CA …DKA †nA dN

…1†

and db ˆ CB …DKB †nB dN

…2†

where N is the number of loading cycles, CA, CB, nA, and nB are material constants, and DKA and DKB are the stressintensity factor ranges at points A and B, respectively. The stress intensity factor was estimated using the Newman± Raju K-solutions [15]. Eqs. (1) and (2) can be combined and given in incremental form C Da ˆ A CB

0308-0161/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0308-016 1(02)00022-4

! DKAnA Db DKBnB

…3†

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P.S. Song et al. / International Journal of Pressure Vessels and Piping 79 (2002) 273±278

Fig. 1. Semi-elliptical surface crack in a ®nite plate under tension.

Integrating Eqs. (1) and (2), N can be expressed as Za 1 Nˆ nA da C …DK a0 A A†

…4†

and Nˆ

Zb b0

1 db CB …DKB †nB

…5†

where n and E are Poisson's ratio and Young's modulus, respectively. To assess the growth of a surface crack, this work extended Sih's through-thickness growth equation to predict the growth behaviour of a surface crack in the crack depth and surface length directions. The extension was performed by transforming Eq. (6) into the following expressions da ˆ C A …DSA †mA dN

…11†

2.2. Sih's energy approach

and

Sih [16] postulated a relationship between the crack propagation rate da/dN and the strain energy density factor range DS for fatigue crack growth. The relationship is de®ned as

db ˆ C B …DSB †mB …12† dN where C A ; C B ; mA and mB are material constants. For the increment in crack depth, Eqs. (11) and (12) can be combined to give ! A C A DSm A Db …13† Da ˆ  m C B DSB B

da m  ˆ C…DS† dN

…6†

where m, C are constants and DS is DS ˆ Smax 2 Smin

…7†

where Smax and Smin are, respectively, the maximum and minimum values of the strain energy density factors. For Mode I fatigue crack growth, DS is given by   DS ˆ a11 …u† KI2 max 2 KI2 min ˆ a11 …u†…1 2 R2 †KI2 max …8† where R is the stress ratio, u is the direction of crack propagation, and KI max and KI min are the maximum and minimum stress intensity factors. For the direction of crack propagation at u ˆ 08; the coef®cient a11 is de®ned as a11

…1 2 2n†…1 1 n† ˆ 2Ep

for plane strain

…9†

and a11 ˆ

…1 2 n† 2Ep

for plane stress

…10†

Integrating Eqs. (11) and (12), N can be expressed as Za 1 Nˆ …14† mA da  a0 C A DSA and Nˆ

Zb b0

1 db B C B DSm B

…15†

3. Experimental procedures The material used for the current study was 2024-T4 aluminium alloy. The chemical composition and mechanical properties of the material are listed in Tables 1 and 2. Surface-cracked specimens were taken from this alloy in the L±T orientation and prepared according to ASTM-E740-80 [17]. The thickness t of specimens was 9.65 mm. The surface notch having an initial depth of a0 ˆ 2:42 mm and

P.S. Song et al. / International Journal of Pressure Vessels and Piping 79 (2002) 273±278

275

Table 1 Chemical composition of 2024-T4 Al-alloy (wt%) Si

Fe

Cu

Mn

Mg

Cr

Zn

Ti

0.5

0.5

3.8

0.3

1.2

0.1

0.25

0.15

Table 2 Mechanical properties of 2024-T4 Al-alloy Yield strength (MPa)

Ultimate strength (MPa)

Young's modulus (GPa)

Elongation (%)

324.6

462.5

73.8

20

an initial length of b0 ˆ 2:54 mm was produced using an electric discharge machine. Fatigue tests were conducted at room temperature using a closed-loop electro-hydraulic machine. Constant amplitude loading was applied with a sinusoidal waveform at a stress ratio of R ˆ 0:1 and a frequency of 5 Hz. Surface length was monitored with a travelling microscope. For crack-depth measurement, a test specimen was fatigued to a speci®ed number of loading cycles then the specimen was broken apart to observe the associated crack front. For the measurements, a total of 18 specimens were used. 4. Results and discussion 4.1. Crack growth behaviour Fig. 2 shows the measured responses of crack depth a and surface length b to the number N of loading cycles. The b±N curve lies above the a±N curve, indicating that the crack growth rate was higher in the length direction than in the depth direction. The higher rate in the length-direction occurred because the crack driving force DKB remained higher than DKA throughout the crack growth process. The behaviour of the ratio of DKA to DKB throughout the growth process is given in Fig. 3. This shows that the ratio varied with measured crack depth and stayed below unity. Fig. 4 shows the crack growth rate against the stress intensity factor range in both depth and length directions. A straight-line relationship was found in each direction and the accompanying Paris equations are da=dN ˆ 5:83 £ 1026 …DKA †1:20

…16†

db=dN ˆ 7:24 £ 1027 …DKB †1:96

…17†

In Fig. 4, the intersection ofpthe  two straight lines occurred at a DK value of 16.4 MPa m; and a corresponding crack 24 growth rate of about p 1.4 £ 10 mm/cycle. At DK-values below 16.4 MPa m; the crack growth rate in the length direction was slightly lower than that in the depth direction. This is because plane stress conditions prevailed in the

Fig. 2. a±N and b±N curves.

length direction [18]. This causes large crack-tip plastic zones, enhancing the existing crack closure effect, reducing crack driving force, and thuspretarding crack propagation. At  DK-values above 16.4 MPa m; the crack growth rate in the depth direction was lower. The lower growth rate is possibly because the crack resistance was more pronounced when the surface crack travelled towards the back surface of the specimen [19]. Fig. 5 indicates the relationships between fatigue crack growth rate and strain energy density factor range in both directions. In each direction, the growth-rate data points fell nearly on a straight line, and the associated Sih's equations are da=dN ˆ 1:31 £ 1023 …DSA †0:59

…18†

and db=dN ˆ 3:35 £ 1023 …DSB †0:94

…19†

where DSA and DSB denote the strain energy density factor ranges for plane strain and plane stress, respectively. 4.2. Crack shape development In inspections of structures it is relatively easy to measure the surface crack length. The primary purpose of crack shape development theory is therefore to predict the crack depth for a given measured surface crack length. In these experiments, measurements were made of the surface crack length development and predictions were then made for the crack depth using Eqs. (3) and (13). The prediction processes were repeated until the predicted crack depths reached a size of about 8.8 mm. Fig. 6 shows that the generalized Sih's approach prediction matched the measured crack depth as did the Paris prediction. To quantify the differences between the measured depths amea and predicted depths apre, this research employed statistical methods [20]. First, the prediction ratio Xi ˆ a…pre†i =a…mea†i was de®ned.

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Fig. 3. Variation of stress intensity factor ratio DKA/DKB with crack depth.

Then the prediction ratios of the 18 surface cracked specimens were calculated. Next the mean value X of the ratios of the 18 specimens was calculated. The specimens had a mean value X of 0.98 for the Paris prediction and 0.99 for the generalized Sih's prediction. The standard deviation for the underlying population is given by  0:5 1 X  2 Sˆ …20† …Xi 2 X† n21 where n denotes the number of specimens. The standard deviation S was 0.025 for the Paris prediction and 0.024 for the generalized Sih's approach prediction, indicating that the generalized Sih's approach predicted the depth more accurately than the Paris equation. Fig. 7 shows the variations of measured and predicted aspect ratios, a/b, with the crack depth ratio, a/t. The

Fig. 5. Relationships between the fatigue crack growth rate and the strain energy density factor range in the depth and length directions.

measured aspect ratio a/b had an initial value of 0.95, increased to a maximum value of 0.97 at an a/t value of 0.31, and then fell in an approximately linear manner to a value of 0.64 at the a/t value of 0.92. The preceding a/b behaviour can be expressed in term of a/t, as a=b ˆ 20:48…a=t†2 1 0:048 …a=t† 1 0:99

…21†

Given the measured surface length b, Eq. (21) can predict the corresponding crack depth a and the accompanying crack shape. Furthermore, Fig. 7 shows that the generalized Sih's approach reproduced the measured crack shape as accurately as the Paris equation. 4.3. Fatigue life predictions Fig. 8 shows the measured and predicted a±N curves. The

Fig. 4. Crack growth rate versus stress intensity factor range in the depth and length directions.

Fig. 6. Comparison of measured and predicted crack depth.

P.S. Song et al. / International Journal of Pressure Vessels and Piping 79 (2002) 273±278

Fig. 7. Comparison of measured and predicted shape variations.

predicted N-values were computed using Eqs. (4) and (14). The ®gure indicates that as the crack extended from the initial depth of 2.42 mm to the ultimate depth of 8.79 mm, the measured number of loading cycles, Nmea, was 24,060 cycles compared to predicted values Npre of 24,222 and 24,011 cycles for Paris and generalized Sih's predictions, …N 2N † respectively. The errors, preNmea mea ; in the predicted values were below 1 and 0.5% for Paris and generalized Sih's predictions. Fig. 9 presents the measured and predicted b± N curves. The predictions of fatigue life N were made using Eqs. (5) and (15). As shown, for the growth interval between the surface lengths of 2.54 and 12.31 mm, the measured number of loading cycles was 24,060 and the predicted number of loading cycles was 24,369 and 24,201 for Paris and generalized Sih's approach predictions, respectively.

277

Fig. 9. Comparison of measured and predicted crack growth in the surface length direction.

The errors in the predicted values were below 2 and 1% for Paris and generalized Sih's approach predictions, indicating that the generalized Sih's approach also enabled accurate predictions of fatigue life in the length direction. Comparisons between the errors of generalized Sih's approach predictions indicate that the generalized approach recorded better accuracy in the depth direction than in the length direction. 5. Conclusions 1. A generalized Sih's energy approach can establish the growth behaviours of surface cracks in a 2024-T4 aluminum alloy in the depth and length directions. 2. A model was established using the measured surface length to predict the crack shape development. 3. The generalized Sih's energy approach predicted closely the depth and fatigue life of the semi-elliptical surface crack, and the predictions compared well with the predictions of the Paris equation.

References

Fig. 8. Comparison of measured and predicted crack growth in the crack depth direction.

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