Crack arrest behaviour and a proposed model

Int J Fatigue 13 No 5 (1991) pp 411-416

Crack arrest b e h a v i o u r and a proposed m o d e l X. D. Zhang and Y. J. Song

The corrosion fatigue propagation behaviours and mechanisms of quenched and tempered AlSl 4340 and 4135 steels have been systematically investigated in 3.5% NaCI spray and in 3.5% NaCI solution. The crack closure for AlSl 4135 steel has been studied in 3.5% NaCI spray and in air. It is shown that there is a great crack arrest tendency in 3.5% NaCI spray but no crack arrest in 3.5% NaCI solution. The crack closure effect for AlSl 4135 steel is higher in 3.5% NaCI spray than in air because of thick oxide deposits within the crack. Through detailed analyses and SEM fractographic examination, it is found that the dominant crack growth mechanism is crack tip anodic dissolution in 3.5% NaCI spray but is hydrogen embrittlement in 3.5% NaCI solution. In this paper, a crack closure blunting model is presented for crack arrest. It is proposed that crack arrest is caused by the combined effects of crack closure and crack tip blunting. The crack arrest tendencies and corrosion fatigue parameters such as frequency, stress ratio and crack tip anodic current density can be predicted and described qualitatively by this model.

Key words: crack arrest; crack tip blunting

It is generally considered that seawater and other aggressive environments accelerate fatigue crack growth. However, both our experimental results and other investigations 1-7 have demonstrated that corrosion fatigue crack growth retardation and arrest can occur in certain material-environment systems. Many hypotheses and explanations, which can be classified into three major aspects such as crack deflection and branching, crack tip blunting as well as corrosion product wedging, have been proposed to account for corrosion fatigue crack arrest phenomena. Crack deflection and branching, which generally exists in corrosion fatigue dominated by hydrogen embrittlement, 3,1° can reduce the effective stress intensity factor range ~¢(efr, and therefore retard crack growth. However, whether or not crack deflection and branching could cause crack growth to arrest completely is a controversial subject. Crack tip blunting is a very important parameter influencing crack arrest. Tomkins 5 proposed that anodic dissolution at a crack tip can modify the crack tip geometry and when anodic dissolution gives a larger effective opening the crack would lose its ability to reshape sufficiently on unloading and would become a notch. Radon et al 6 showed that crack tip blunting seems to be dependent upon the crack growth rate d a / d N in mild steel; crack blunting and subsequent arrest do not occur above a certain critical value (da/dN)b. This process was explained by a combination of dissolution and the mechanical effects of fatigue. The dissolution process contributes mainly by increasing the radius of curvature at the crack tip, blunting the crack therefore and finally leading to its arrest. In order to determine the maximal value of Ml" for crack retardation

and arrest, constant-~K tests were run by van der Velden et al 1° in mild steel at R = 0.1. In oxygen-saturated seawater at a frequency of 10 Hz some retardation and arrest were observed at M¢ = 30 MPa Vmm, but the effect was absent at 35 MPa ~mm. For air-saturated seawater the equivalent values were 20 and 25 MPa V~m, respectively. It is implied that there exists a critical AK value for crack arrest and that oxygen accelerates the crack arrest. The relationship between the critical AiK value and the external variables was not considered by the authors. Corrosion product wedging causes a strong crack closure effect, which increases the minimal value of K and hence reduces the effective stress intensity factor range Mqerr. This oxide-induced crack closure effect, when the oxide thickness within the crack is larger than the crack tip opening displacement, can enhance the threshold stress intensity factor range ~¢~th greatly, 7,s and can make the crack retard and arrest. 9.10 This paper will demonstrate that the mechanisms responsible for corrosion fatigue crack growth in 3.5% NaC1 solution and 3.5% NaCl spray are hydrogen embrittlement and anodic dissolution, respectively. A new model that is based on the combination of oxide-induced crack closure and anodic dissolution is proposed to explain the corrosion fatigue arrest in oxygen-sufficient environments such as 3.5% NaCl spray.

Experimental procedures The chemical compositions of 40CrNiMo and 35CrMo, namely AISI 4340 and AISI 4135, are given in Table 1. The two steels are fully martensitic following oil quenching from

0142-1123/91/050411-06 (~ 1991 Butterworth-Heinemann Ltd Int J Fatigue September 1991

411

T a b l e 1. C h e m i c a l c o m p o s i t i o n (in %)

of t h e t e s t e d steels S

Material AlSl 4135 AISI 4340

C

Si

Mn

P

S

Cr

Ni

T

Mo

0.36 0.30 0.52 0.017 0.005 0.95 0.00 0.18 0.39 0.29 0.59 0.019 0.007 0.72 1.50 0.19

850 °C prior to tempering (2 h) at 560 °C. The ultimate tensile strengths are 1070 MPa and 1060 MPa, respectively. Two different standard three-point specimens are used in the experiments. The 12 × 24 × 100 mm 3 specimens, with a 3 mm notch and a 1.5 mm precrack, are only used for the 0.1 Hz corrosion fatigue. The 25 × 50 × 200 mm ~ specimens, having a 4 mm notch and a 3 mm precrack, are examined in 5.5 Hz corrosion fatigue and stress corrosion cracking tests. Precracking is performed in ambient air by using successive decreases of about 10% in load. A closed loop and hydraulically activated machine with sinusoidal tension-tension loading and a stress ratio of one third is employed for all the corrosion fatigue tests. The crack length is measured with an optical microscope on each side of the specimen. A self-made spray apparatus is attached to the machine for 3.5% NaC1 spray corrosion fatigue tests. A six-hour spraying time and six-hour stop time periodicity is used for the spray. During the six-hour stop times, the relative humidity is well above 85%. While in the six-hour spraying period the mean spraying fall rate is 1-2 ml/80 cm 2 h. Crack closure measurements are made by using a series of strain gauges located on the flanks of the specimens (Fig. 1). When the growing crack tip is in close proximity to the middle of a strain gauge the strain gauge is used for recording the change of the strain at the crack tip. Load-displacement P-~ curves are recorded in the unloading process. Crack closure is determined from the non-linear behaviour of the P-5 curve. As shown in Fig. 2 the load corresponding to the point S is taken as the opening load (Pop), therefore the actual values of APeff and ~rl'~f~ are calculated.

Fig. 2 Schematic diagram of the unloading P-5 curve in the test

The crack growth rate is higher in 3.5% NaC1 solution than in 3.5% NaC1 spray when ~kK is less than 19 MPa ~/m. An instant brittle fracture in 3.5% NaCI solution occurred when ~lK reached 36 MPa Vmm. The specimens still have enough strength and plasticity at &K = 36 MPa ~mm in 3.5% NaC1 spray. There is a valley on the da/dN-AK curve in 3.5% NaCI spray and a crack growth retardation and arrest tendencies appeared. The anomaly of the crack growth behaviour at a frequency of 5.5 Hz in the 3.5% NaCI spray is illustrated in Fig. 4. When the initially applied &K is smaller than the initial &K shown in Fig. 3, the oxygen-induced crack closure effects and crack tip blunting decrease the effective stress intensity factor range to such a level that make the crack totally arrest, as shown in curve 1 and curve 2 of Fig. 4. There are no corrosion products within the crack and the specimens are

I

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10 .3 -

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~

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3.5°/oNaCI Solution 3.5%NaCI Spray

AA A~

Results

#

104

S t e s s c o r r o s i o n cracking The stress corrosion cracking (SCC) tests are employed for the two steels in 3.5% NaC1 spray. Many specimens are used but no definite Kt~c,. data are obtained. Namely, the two steels are not sensitive to SCC in 3.5% NaC1 spray.

I

f

.9o E vE Z x x

Corrosion fatigue The crack growth behaviours at a frequency of 5.5 Hz in 3.5% NaC1 spray and in 3.5% NaC1 solution have been systematically investigated by the authors. The results of the corrosion fatigue crack growth tests in 3.5% NaC1 spray and in 3.5% NaC1 solution at 5.5 Hz frequency are given in Fig. 3.

4'

X

1 0 "~

x

10 .6

I

I

I

I

4

6

10

20

I

I

40 60

I

I

100

200

I ~ Strain gauge

AK (MN/rn~2) Fig. 1 Schematic diagram of the strain gauge position at the tested specimen

412

Fig. 3 Comparison of the corrosion fatigue behaviours of AISI 4135 steel tempered at 560°C in 3.5% NaCI solution and in 3.5% NaCI spray

Int J F a t i g u e S e p t e m b e r 1991

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ld 3

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1¢ 3 2. 1A-..~

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20 40 6 0 1 0 0 2 0 0

A K (MN/rn3' Fig. 4 Schematic illustration of anomalous crack propagation behaviours in 3.5% NaCI spray

clean before the test. Once the specimen is put into the corrosive environments and is under test, the crack arrest tendencies are negligible at the beginning of the test because it takes time for the corrosion products to accumulate within the crack. Therefore, the initial crack growth rate is the largest, which is the same as the pure mechanical fatigue crack growth rate, as shown in point A of curve 1 and point C of curve 3. With the time increased, the corrosion products begin to accumulate within the crack, the crack closure and crack arrest tendencies increase. Accordingly, the crack growth rate begins to diminish as shown in AB of curve 1 and CD of curve 3. When the initial value of M( is relatively small, as shown in curves 1 and 2, the mechanical effect is not enough to make the crack continue to grow and total crack arrest occurs. When the initial value of &K is large, as shown in curve 3 of Fig. 4, the mechanical effect makes the crack continue to grow. In the stage CD of curve 3, the decrease of the crack growth rate is caused by crack arrest tendencies. In the stage DE of curve 3 the mechanical effect dominates the crack propagation processes. It is noted that at point D the crack still continues to grow but the crack growth rate is smaller than at point C. In addition, the above experimental results are in agreement with the constant-~tK results of van der Velden et al, as we mentioned in the introductory section. 1° From the above illustration it is revealed that the anomalous crack propagation behaviour observed in 3.5% NaCI spray is caused by a combination of mechanical effects and crack arrest tendencies rather than a transient phenomenon following the start of the test. The detailed analyses of the anomalous crack propagation behaviour in 3.5% NaCI spray will be presented in the discussion section. The corrosion fatigue crack growth tests at the frequency

Int J Fatigue September 1991

of 0.1 Hz in 3.5% NaCI solution 2s and in 3.5% NaC1 spray have been systematically studied for AISI 4340 and 4135 steels. It is shown that in 3.5% NaCI solution the d a / d N - ~ curves can be obtained easily at a frequency of 0.1 Hz. 28 Dozens of crack growth tests in 3.5% NaCI spray at a frequency of 0.1 Hz have been done by the author using the same machine and following the same test procedures as in the 3.5% NaCI solution. It is clearly revealed that the da/dN-&Kcurve in 3.5% NaC1 spray at a frequency of 0.1 Hz can not be obtained because of serious crack growth retardation and arrest. The anomaly of the crack growth behaviour in 3.5% NaCI spray at a frequency of 0. I Hz is genuine rather than an artefact associated with the start of the test. This will be explained in the following discussion. Metallographic sectioning of specimens fatigued in 3.5% NaCI spray at frequencies of 0.1 Hz and 5.5 Hz revealed that there was a lot of serious crack blunting caused by anodic dissolution near the crack tip, as shown in Fig. 5. The detailed fracture surface morphologies of AISI 4135 steel specimens tested in 3.5% NaCI spray and in 3.5% NaCI solution have been examined by scanning electron microscopy (SEM), as shown in Fig. 6. Fractography indicates that throughout the range observed above the threshold, crack growth is taken on the stage II ductile striation mechanism in 3.5% NaCI spray (Figs 6(a, b)), while in 3.5% NaC1 solution, the crack growth is characterized by intergranular separation and by transgranular cleavage (Figs 6(c, d)).

Crack closure effect The results of crack closure measurements in 3.5% NaCI spray and in air for AISI 4135 steel are shown in Fig. 7. It is revealed that the crack closure effect is much higher in 3.5% NaCI spray than in air. Through crack surface examination it is found that the crack surfaces in 3.5% NaCI spray are always covered with a thick layer of corrosion products, whereas the crack surfaces in air are clean. Because the plasticity-induced and roughness-induced crack closure are diminished and negligible for stage-II crack growth it is concluded that the crack closure in 3.5% NaC1 spray is caused by the thick oxide deposits within the crack.

Discussion Different corrosion fatigue propagation mechanisms According to basic electrochemistry theory, 25 deformed areas in metals and alloys are anodic to undeformed areas. If there is a tensile stress concentration near the crack tip then the crack tip is the anodic region and the inner surface of

a|

by

Fig. 5 Bluhting crack tip morphology of the tested steels: (a) AISI 4340, 5.5 Hz, 500x; (b) AISI 4340, 0.1 Hz, 500×

413

A

Fig. 6 SEM F r a c t o g r a p h y o f AISI 4135 steel: (a) &K = 12 MPa X/m, 3.5% NaCI spray, 500x; (b) &K = 33 MPa X/m, 3.5% NaCI spray, 500x; (c) &K = 16 MPa ~/m, 3.5% NaCI solution, 500x; (d) &K = 37 MPa ~/m, 3.5% NaCI solution, 300x

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1.0

A

E E

Spray Air

-

.--- 0.8 0.6

Fe--> Fe 2+ + 2e

A,

A,

takes place, while at anodic regions the following alternanve reactions are possible: H 2 0 + ½02 + 2e--> 2 O H -

A

"~ 0.4

l

"O

S

3

H

3X

X

X

B

0.2 0

15 t~K(MN/m~2)

20

25

Fig. 7 Results of crack closure measurement of AlSl 4135 steel tempered at 560 °C in 3.5% NaCI spray and in air

the crack is the cathodic region; therefore, electrochemical corrosion will occur within the crack enclave. The corrosion reactions occurring at the local anodes and cathodes may be expressed as follows. 26 At the anodic region, the reaction

414

+ e--> Hadsorbed

2Hadsorbed ---->H2

I

0.0

(2)

or

A

Z

(1)

(3)

The hydrogen reaction (reaction 3) and the oxygen reduction (reaction 2) play different roles in different environmental conditions. In 3.5% NaCI solution oxygen diffusion to the crack tip is very slow and difficult, whereas hydrogen ions can easily enter the inner surface of the crack so that pH of the solution near the crack tip is usually lower than the average value of the pH of the solution. Therefore atomic hydrogen can be easily produced, enter the metal lattice and cause hydrogen embrittlement. In 3.5% NaCI spray there is only a very thin layer of 3.5% NaC1 solution on the specimens, oxygen can easily diffuse into the crack and, accordingly, the anodic dissolution dominated. Many deposits were accumulated within the crack by the following reactions: 27

Int J Fatigue September 1991

Fe z+ + 2 O H - - - - ) Fe(OH)2 4Fe(OH)2 + 02 + 2 H 2 0 ~ 4Fe(OH)3

°T

(4)

2Fe(OH)3 + Fe(OH)2--* Fe304 + 4 H 2 0 The above chemical reactions coincided with the present experimental results. Whether anodic dissolution or hydrogen embrittlement mechanisms dominated in corrosion fatigue cannot be determined accurately. It is fortunate that fractographic analyses do provide supporting evidence for different fracture mechanisms. Tomkins 11 showed that under the control of an anodic dissolution mechanism crack growth is the result of a stageII ductile striation mechanism above the threshold. Hydrogen embrittlement is always characterized by intergranular separation and by transgranular cleavage and quasicleavage. 12'13 Many investigators take the fractographic examination as the main tool to judge which mechanism is dominating in the corrosion fatigue crack propagation. 14-17 These investigations are in agreement with our results as shown in Fig. 6. According to the above analyses it is concluded that, for high-strength AISI 4340 and 4135 steels, corrosion fatigue crack propagation is controlled by the anodic dissolution mechanism in 3.5% NaCI spray and by hydrogen embrittlement in 3.5% NaCI solution.

A crack closure-blunting model The crack growth arrest is caused by many parameters. The percentage of oxygen within the crack is a key factor in influencing crack arrest. Oxygen within the crack can enhance the anodic dissolution rate at the crack tip, therefore making the crack blunt. Bristoll and Roeleveld 18 and Johnson et aP 9 proposed that oxygen within the crack causes the crack blunting and arrest in structural steel during fatigue at 0.1 Hz under tidal immersion conditions when the crack growth rate is comparatively low (< 10 -s ram/cycle). This is in agreement with the corrosion fatigue results in 3.5 % NaCI spray as shown in Fig. 3. On the other hand, oxygen can increase oxide deposits on the inner surface of the crack and a significant oxide-induced crack closure could occur. The frequency f and stress ratio R also play important roles in corrosion fatigue. In a certain range, decreases of f and R can enhance crack arrest tendencies. 18'19 It has been demonstrated that the oxide-induced crack closure effect is higher in oxygen-sufficient 3.5% NaCI spray than in 3.5% NaC1 solution and in air. The crack arrest in 3.5% NaCI spray is always accompanied by crack tip blunting. From these facts and the above analyses, a fundamental hypothesis could be proposed: under an oxygen-sufficient environment, oxygen-induced crack closure effects make the effective stress intensity factor range AKeff and crack growth rate decrease; accordingly, anodic dissolution at the crack tip has enough time to make the crack tip blunt, and crack arrest occurs. In other words, whether or not crack arrest could occur in an oxygen-sufficient environment depends upon the competition of crack growth rate and crack tip blunting rate. When the ratio of crack growth rate to crack tip blunting rate is less than a constant, crack tip blunting and crack arrest could appear. As shown in Fig. 8 the crack tip could be conveniently represented mathematically by an elliptical or hyperbolic cylinder, void of material, in which the radius of curvature at the tip is small in comparison to the major dimensions of the void. 2° The condition for crack arrest to occur can be expressed by

Int J Fatigue September 1991

"lUP-

Fig. 8

Schematic diagram of the crack tip configuration

a,'p ~< c,

(5)

where 4 is the crack growth rate, 15 is the rate of increase of the crack tip curvature radius and CI is an experimental constant.

The crack g r o w t h rate The crack closure coefficient U is given by U = AKm~x -- AK°p -- AKeff AKmax -- AKmi n

(6)

where Kmax and Kmi, are the maximal and minimal stress intensity factor values, respectively. Elber a1'24 and Schijvez2 have shown that U has an empirical linear relationship only with stress ratio R expressed by Ue = 0.5 + 0.4R for - 0 . 1 ~< R ~< 0.7 and U, = 0.55 + 0.35R + 0.1R 2 for - 1 ~< R ~< 1 respectively. According to these investigations, the relationship between U and R can generally be expressed as U = aR 2 + bR + c

(7)

where a, b and c are positive constants. Combining Equation (6) and the Paris law, the crack growth rate can be given by

a = f d a / d N = fC2(AKe.)" = fC2U"AK"

(8)

where f is frequency, (72 is the Paris coefficient and n is the Paris exponent.

The rate o f increase o f the crack tip curvature radius [~ The rate of increase of the crack tip curvature radius 15 can be expressed by adding 15A and hiE, namely h = hA + hE

(9)

where hA is the rate of increase of the crack tip curvature radius caused by anodic dissolution at the crack tip and hE is the rate of increase of the crack tip curvature radius arising from the crack tip plastic deformation. If the dominant mechanism is crack tip anodic dissolution then hA ~> Pr, hr will be negligible and hence h can be obtained only by hA:23 h = hA = M i a / Z F d

(10)

where ia represents the crack tip anodic current density, M denotes the atomic weight of the dissolving metal with density d, Z is the charge on the solvated metal cation and F is the

415

Faraday constant. Substituting Equations (7), (8) and (10) into Equation (5), one can obtain

References 1.

Suresh, S. Metall Trans 14A (1983) p 2375

2.

Vitek, V. Int J Fract Mech 13 (1977) p 481

(11 )

3.

Tu, L. K. L. and Seth, B. B. J Testing Evaluation 6 (1978) p 66

where C = (Cl/C2) l/~ and is a constant. If the left-hand side of Equation (11) equals the right-hand side, a critical nominal stress intensity factor range can be given from Equation (11), namely

4.

Carter, C. S. Metall Trans 1A (1970) p 1551

5.

Tomkins, B. Met Sci 13 (July 1979) p 387

6.

Radon, J. C. et al Int J Fract 12 (1976) p 467

7.

Suresh, S., Zamiski, G. F. and Ritchie, R. O. Metall Trans 12A (1981) p 1435

A K <~ C l M i a l 1/" aR z + bR + c ~ZFdf]

C ( M i a ] '/" M(cr~ = aR 2 + b l ( + c \-ZF~lf]

(12)

AKcri is an important criterion and controlling parameter of crack arrest. When AK < AK~ri crack arrest can occur; whereas when AK > AKcri no crack arrest could exist. 8Kcri itself is also a symbol of crack arrest tendency. The greater the AKcri value, the higher the crack arrest tendency. From Equation (12) it is indicated that with f and R increased, AKcri diminished, and therefore crack arrest tendency decreased, or vice versa. This implication is in agreement with the results of Bristoll and Roeleveld 18 and Johnson et al.19 Increasing ia can enhance crack arrest tendency. Equation (12) explains adequately why there exists a critical crack growth rate value and why a certain AK value for crack arrest fatigued at 5.5 Hz frequency in 3.5% NaC1 spray appears. It is noted that the above theoretical model for crack retardation and arrest is of limited value for a complete quantitative examination unless all the variables involved in Equation (12) can be obtained. No attempts to monitor crack tip anodic current density appear to have been made so far and it seems, therefore, that if a complete quantitative examination of crack arrest is to be obtained, experiments designed to achieve crack tip anodic current density during corrosion fatigue are desirable.

8.

Suresh, S. Scripta Metall 16 (1982) p 995

9.

Normark, G. E. and Fricke, W. G. J Testing Evaluation 6 (1978) p 301

10.

van der Velden, R. etalASTMSTP801 (American Society for Testing and Materials) pp 64-80

11.

Tomkins, B. Influence of Environment on Fatigue (Mechanical Engineering Publications Limited for the Institution of Mechanical Engineers, London, 1977) p 111

12.

Chu, W. Y., Hsiao, C. M. and Li, S. Q. Acta Metall Sinica 17 (1981) p 10

13.

Xu Jian et al Corrosion Metallography and Corrosion Resistant Metallic Materials (Publishing House of Science and Technology of Zhejiang, Zhejian9, China, 1981) p 124

14.

Nelson, H. G., Williams, D. P. and Tetelman, A. S. Metall Trans 2 (1971) p 953 Beachem, C, D. Metall Trans 3 (1972) p 437

15. 16.

Gao, M., Lu, M. and Wei, R. P. Metal/ Trans 15 {1984) p 735

17.

Tong Zhi-Shen et al Corrosion 41 (1985) p 121 Bristoll, P. and Roeleveld, J, A. Proc of European Offshore Steels Research Seminar, 1980 (The Welding Institute, Cambridge, UK, 1980) pp VI/p 18-VI/p 18-10 Johnson, R, et al ibid pp VI/p 15-VI/p 15-15 Geager, M. and Paris, P. C. Int J Fract Mech 3 (1967) p 247 Elber, W. 'Damage tolerance in aircraft structure' ASTM STP 486 (American Society for Testing and Materials, 1971) pp 230-242 Schijve, Jr. Department of Aerospace Engineering Memorandum M-336 (Delft University, Delft, The Netherlands, 1979)

18.

19. 20.

Conclusions 1)

2)

3)

4)

416

In quenched and tempered AISI 4340 and 4135 steels there are significant crack arrest tendencies in 3.5% NaCI spray fatigued at frequencies of 0.1 Hz and 5.5 Hz but no crack arrest is found in 3.5% NaCl solution and in air. In a certain range of nominal AK these crack arrest tendencies increased as the frequencies decreased. For quenched and tempered AISI 4135 steel fatigued at 5.5 Hz frequency, the crack closure effect is higher in 3.5% NaCI spray than in 3.5% NaCI solution and in air because of thick oxide deposits within the crack. It is revealed that the dominant corrosion fatigue crack propagation mechanism is crack tip anodic dissolution in 3.5% NaCI spray but hydrogen embrittlement in 3.5% NaCl solution according to theoretical analyses and SEM examination. A crack closure blunting model is presented for crack arrest. The model predicts that crack arrest tendencies are determined by a critical nominal stress intensity factor range AKcri. The marked influences of frequency, stress ratio and crack tip anodic current density can be explained by the model as well. The prediction of this model is in qualitative agreement with the experimental data reported in the open literature.

21.

22.

23.

Congleton, Jo and Craig, I. H., in Corrosion Processes Ed R. N. Parkins (Applied Science Publishers, London, 1982) pp 209-269

24.

Elber, W. Eng Fract Mech 2 (1970) p 37

25.

Evans, U. R. The Corrosion and Oxidation of Metals: Scientific Principles and Practical Applications (Edward Arnold, London, 1960) p 681 Hughes, P. C. et al J Iron Steel Inst 203 (July 1965) p 728

26. 27.

Huan Suju J Xi'an Jiaotong University 18 (1984) p 49

28.

Quan Gaofeng Master's Thesis(Xi'an Jiaotong University, Xi'an, China, 1986)

Authors X. D. Zhang is with the Center for Intelligent Material Systems and Structures, Virginia Polytechnic Institute and State University, Blacksburg, VA 240601, USA. Y. J. Song is at the Department of Materials Science and Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China. X. D. Zhang was formerly at Xi'an Jiaotong University.

Int J F a t i g u e S e p t e m b e r

1991