- Email: [email protected]

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Fatigue behavior of aluminum alloys under biaxial loading E.U. Lee ⇑, R.E. Taylor Naval Air Warfare Center Aircraft Division, Patuxent River, MD, USA

a r t i c l e

i n f o

Article history: Available online 19 November 2010 Keywords: Biaxiality Fatigue life Crack path Fatigue crack growth In-phase and out-of-phase loading

a b s t r a c t The biaxiality effect, especially the effect of non-singular stress cycling, on the fatigue behavior was studied, employing cruciform specimens of aluminum alloys 1100-H14 and 7075-T651. The specimens, containing a transverse or a 45o inclined center notch, were subjected to in-phase (IP) or 100% out-of-phase (hereinafter referred to as ‘‘out-ofphase or OP’’) loading of stress ratio 0.1 in air. The biaxiality ratio k ranged from 0 to 1.5, and 3 levels of stress were applied. It was observed that: (1) at a given k, a lower longitudinal stress induced a longer fatigue life under IP and OP loading, and the fatigue life was longer under IP loading, (2) the fatigue crack path proﬁle was inﬂuenced by k, phase angle (0o or 180o), and initial center notch (transverse or 45o inclined); (3) the fatigue crack path proﬁles, predicted analytically and determined experimentally, had similar features for the specimens with a transverse center notch under IP loading; and (4) the fatigue crack growth rate was lower and the fatigue life longer for a greater k under IP loading, whereas it changed little with change in k under OP loading. These results demonstrate that nonsingular stress cycling affects the biaxial fatigue behavior of aluminum alloys 1100-H14 and 7065-T651under IP and OP loading. Published by Elsevier Ltd.

1. Introduction Metal fatigue has been studied mostly under uniaxial stressing for the fatigue life prediction, design and maintenance of structural components. However, the fatigue-prone portions in structures are subject to substantial levels of biaxial stress. Current methods of fatigue prediction, which generally ignore biaxiality, are often inaccurate. The biaxial stressing arises from geometry, material inhomogeneity, and loading in different directions with different frequencies and/or different phases. The effects of biaxial stressing on fatigue have been investigated theoretically [1,2] and experimentally [3,4] by many researchers, and various conclusions, some contradictory, have been drawn. Elastic analyses indicate that biaxial stresses would produce only a second-order effect on the fracture properties of ideally elastic materials. Since real materials exhibit plastic behavior near the crack tip and the plastic deformations depend on the entire stress state and history, one cannot assure that loading parallel to the crack will not affect the conditions near the crack tip and alter the fatigue and fracture behavior of the material. There is considerable evidence in the literature to demonstrate that the non-singular stress, parallel to a crack, has an inﬂuence on the fatigue crack growth, even though it does not contribute to the stress intensity factor [5]. This has been observed in aluminum alloys [6], steels [4] and polymers [7]. The crack-tip plastic zone size is inﬂuenced by the biaxial stress state, and this may be demonstrated to affect the fatigue crack growth. Furthermore, it has been reported that the tensile transverse stress can increase [8–10], reduce [2,6,11], or have little effect [12] on fatigue crack growth rate, and can cause instability of the crack path [13].

⇑ Corresponding author. Tel.: +301 342 8069. E-mail address: [email protected] (E.U. Lee). 0013-7944/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.engfracmech.2010.11.005

1556

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

Nomenclature

a k

r kr

rhh r, h x, y a KI KII N R IP OP

crack angle biaxiality ratio longitudinal stress transverse stress tangential stress component cylindrical polar coordinates of a point with respect to a crack tip Cartesian coordinates of a point with respect to a crack tip a half crack length Mode I stress intensity factor Mode II stress intensity factor fatigue life stress ratio in-phase out-of-phase

A study has been conducted to clarify the effect of biaxial stressing, including non-singular stress cycling parallel to the crack plane, on the fatigue crack path, growth and life. The results with aluminum alloys are discussed in this paper.

2. Experimental procedure 2.1. Material and specimen As the specimen materials, 2 mm thick sheets of aluminum alloys, 1100-H14 and 7075-T651, were selected. Their chemical compositions and mechanical properties are shown in Tables 1 and 2, respectively. From these sheets, cruciform specimens were machined to have the overall length or width of 393 mm, including the grip areas of the loading arms. A sketch of the specimen is shown in Fig. 1. The vertical arms were in the longitudinal (or rolling) direction and the horizontal ones in the transverse direction of the sheet. Each arm was 127 mm wide and 133 mm long. At the specimen center, a transverse (or horizontal) notch or a 45o inclined one, 38 mm long and 0.25 mm wide, was made by electro-discharge machining. The transverse notch was made in the 1100-H14 specimen and the 45o inclined one in the 7075-T651 specimen. Subsequently, a precrack was made under cyclic biaxial loading until its length reached 1 mm from each end of the central notch. 2.2. Biaxial fatigue test The biaxial fatigue test was conducted in a MTS Model 793.10 Multiaxial Purpose Test-Ware with two pairs of servohydraulic actuators and two pairs of load cells, arranged perpendicular to each other on a horizontal plane in a rigid frame. It was capable of static and cyclic biaxial loading in vertical and horizontal directions, separately or simultaneously. Tensile or compressive loads could be applied to each pair of the arms, developing a biaxial stress ﬁeld in the working section. The cyclic biaxial loading, IP or OP, was done at various longitudinal stresses r and biaxiality ratios k, stress ratio R = 0.1 and loading frequency 15 Hz in air. The growing crack length was measured by means of DC potential drop method. When the crack length reached 140 mm, it was deﬁned that the specimen was failed by fatigue. The fractograph was examined in a scanning electron microscope, JEOL SEM JSM-6460LV, operated at an accelerating voltage of 20 kV.

3. Experimental results The experimental results are divided into two parts: biaxial fatigue behavior of specimen with a transverse notch and that with a 45o inclined notch under IP and OP loading.

Table 1 Chemical compositions of aluminum alloys 1100 and 7075 (wt.%). Al-alloys

Si

Fe

Cu

Mn

Mg

Cr

Zn

Ti

Other

Al

1100 7075

0.11 0.50

0.53 0.7

0.0765 1.2–2.0

0.0039 0.30

0.0010 2.1–2.9

0.0017 0.18–0.4

0.0027 5.1–6.1

0.0011 0.20

0.15 0.15

Balance Balance

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

1557

Table 2 Mechanical properties of aluminum alloys 1100-H14 and 7075-T651. Al-alloys

YS (MPa)

UTS (MPa)

Elongation (%)

1100-H14 7075-T651

129 462

132 524

9 11

393 127

133

133

133

Rolling direction

393

133

127

25R

38

(in mm) (2.3 mm thick)

Fig. 1. Sketch of specimen.

3.1. Biaxial fatigue behavior of 1100-H14 aluminum alloy specimen with a transverse notch Under IP loading, the fatigue crack path was transverse for k 6 1, but began to deviate and proceed towards the radius between the loading arms with k increasing from 1 to 1.5. On the other hand, under OP loading, it was transverse for the all k values employed, 0.5, 1 and 1.5. A summary sketch of the biaxial fatigue crack paths is shown in Fig. 2. Under IP loading, (1) a greater k extended N at a given r, and (2) a greater r and a greater kr shortened N at a given k, as shown in the plots of k vs. N, r vs. N and kr vs. N, Fig. 3. Under OP loading, (1) N was changed little by increasing k at r 6 42.9 MPa but shortened at r 6 51.5 MPa, and (2) a greater r and a greater kr shortened N at a given k, as shown in the plots of k vs. N, r vs. N and kr vs. N, Fig. 4. The fatigue life variations under IP and OP loading are compared at a longitudinal stress r = 51.5 MPa in the plots of k vs. N and kr vs. N, Fig. 5. N was shorter under OP loading, and the difference was greater at a greater k and a greater kr. The variation of fatigue crack growth under IP and OP loading of a given r = 35.7 MPa is shown in the plots of a vs. N, Fig. 6. Under IP loading, the crack growth was faster and the life was shorter for a smaller biaxiality ratio. However, under OP loading, the three curves of a vs. N plot nearly overlap each other, indicating little inﬂuence of k on the fatigue crack

Fig. 2. Sketch of biaxial fatigue crack paths in specimens with a transverse notch.

1

λ

80

35.7 42.9 51.5

1.5

100

0.5 0 1.E+05

1.E+06

1.E+07

Number of Cycle, N

60

1.5 1 0.5

40 20 0 1.E+05

1.E+06

1.E+07

Number of Cycle, N

Transverse Stress,

σ (MPa)

Longitudinal Stress,

Biaxiality Ratio,

2

(MPa)

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

(MPa)

1558

100

λ

80

1.5 1 0.5

60 40 20

0 1.E+05

1.E+06

1.E+07

Number of Cycle, N

σ (MPa)

1.5

80

35.7 42.9

1

51.5

0.5 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

60

(MPa)

100

Longitudinal Stress,

Biaxiality Ratio,

2

λ

40 20 0 1.E+04

100

λ

80

1.5 1 0.5

1.E+05

1.E+06

1.E+07

Number of Cycle, N

Transverse Stress,

(MPa)

Fig. 3. Variation of fatigue life N with biaxiality ratio k, longitudinal stress r and transverse stress kr under in-phase loading in specimen with a transverse notch.

1.5 1 0.5

60 40 20 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

Fig. 4. Variation of fatigue life N with biaxiality ratio k, longitudinal stress r and transverse stress kr under out-of-phase loading in specimen with a transverse notch.

(MPa)

1.5

IP

Transverse Stress,

Biaxiality Ratio,

2

OP 1

0.5

0 1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

100 80

IP

60

OP

40 20 0 1.E+03

1.E+04

Number of Cycle, N

1.E+05

1.E+06

1.E+07

Nymber of Cycle, N

Fig. 5. Comparison of fatigue life N variation with biaxiality ratio k and transverse stress kr under in-phase and out-of-phase loadings of longitudinal stress r = 51.5 MPa in specimens with a transverse notch.

90

90

λ

80

1.5 1

70 60

Half Crack Length, a (mm)

Half Crack Length, a (mm)

100

0.5

50 40 30 20 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

80 70

λ 1.5 1 0.5

60 50 40 30 20 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

Number of Cycle, N

Number of Cycle, N

In-Phase Loading

Out-of-Phase Loading

1.E+07

Fig. 6. Fatigue crack growth under in-phase and out-of-phase cyclic loading of longitudinal stress r = 35.7 MPa at biaxiality ratios k = 0.5, 1 and 1.5 in specimens with a transverse notch.

Half Crack Length, a (mm)

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

1559

90 80

IP

70

OP

60 50 40 30 20 1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N Fig. 7. Comparison of fatigue crack growth under in-phase and out-of-phase loading of longitudinal stress r = 35.7 MPa at biaxiality ratio k = 1 in specimens with a transverse notch.

Fig. 8. SEM fractographs of initial transverse and subsequent deviated portions of a fatigue crack path.

growth. The fatigue crack growths under IP and OP loading are compared at k = 1 in Fig. 7. It is noticeable that the fatigue crack growth was faster and the fatigue life was shorter under OP loading at a given biaxiality ratio. As the sample fractographs, the SEM fractographs for the initial transverse and subsequent deviated portions of a fatigue crack path are shown in Fig. 8 for the case of k = 1.5 and IP loading. They had similar fractographic features: clearly deﬁned ductile striations with secondary cracks along some striations. The noticeable difference was the wider striation and the larger fatigue patch in the deviated portion.

Fig. 9. Sketch of biaxial fatigue crack paths in specimens with a 45o inclined notch.

Biaxiality Ratio,

σ (MPa)

1.5

53.6 69 84.3

1 0.5 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

140

(MPa)

2

λ

120

1.5 1 0.5 0

100 80 60 40 20 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

140

λ

120

1.5 1 0.5 0

100

Transverse Stress,

(MPa)

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

Longitudinal Stress,

1560

80 60 40 20 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

σ (MPa)

1.5

53.6 69

1

84.3

0.5 0 1.E+04

1.E+05

1.E+06

Number of Cycle, N

1.E+07

(MPa)

140 120

λ

100 80

1.5 1 0.5 0

60 40 20 0 1.E+04

1.E+05

1.E+06

1.E+07

Transverse Stress,

Biaxiality Ratio,

2

Longitudinal Stress,

(MPa)

Fig. 10. Variation of fatigue life N with biaxiality ratio k, longitudinal stress r and transverse stress kr under in-phase loading in specimens with a 45o inclined notch.

140 120 100

λ 1.5 1 0.5 0

80 60 40 20 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

Number of Cycle, N

Fig. 11. Variation of fatigue life N with biaxiality ratio k, longitudinal stress r and transverse stress kr under out-of-phase loading in specimens with a 45o inclined notch.

3.2. Biaxial fatigue behavior of 7075-T651 aluminum alloy specimen with a 45o inclined notch Under IP loading, the crack path was vertical for k = 1.5, diagonal along the 45o inclined notch for k = 1, and transverse for k = 0.5 and 0. On the other hand, under OP loading, the crack path was vertical for k = 1.5, and transverse for k = 1, 0.5 and 0. A summary sketch of the crack paths is shown in Fig. 9. Under IP loading, (1) the fatigue life was extended by increasing k from 0 to 1 but shortened by increasing k from 1 to 1.5, resulting in the fatigue life longest at k = 1 at a given r, and (2) a greater r and a greater kr shortened N at a given k, as shown in the plots of k vs. N, r vs. N and kr vs. N, Fig. 10. Under OP loading, (1) N was shortened by increasing k from 1 to 1.5 at a given r, and (2) a greater r and a greater kr shortened N at a given k, as shown in the plots of k vs. N, r vs. N and kr vs. N, Fig. 11. The fatigue life variations under IP and OP loading are compared at r = 53.6 MPa in Fig. 12. The fatigue life was shorter under OP loading, and the difference was greater at greater k and kr.

4. Discussion 4.1. Analysis of crack path in a specimen containing a transverse central notch

1.5

100 80

IP

60

OP

IP OP

1

0.5

0 1.E+04

(MPa)

Biaxiality Ratio,

2

Transverse Stress,

In the maximum stress criterion for brittle fracture [14], the crack is hypothesized to grow along a path normal to the direction of maximum tension. In this case, the component of shear stress on the line of expected crack growth is zero. This

1.E+05

1.E+06

Number of Cycle, N

1.E+07

40 20 0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycle, N

Fig. 12. Comparison of fatigue life N variation with biaxiality ratio k and transverse stress kr under in-phase and out-of-phase loading of longitudinal stress r = 53.6 MPa in specimens with a 45o inclined notch.

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

1561

Fig. 13. A biaxially loaded plate containing a crack.

Table 3 Calculated angle of crack extension. k

h

0.5 1 1.5

0° 0o 56o60

criterion has offered reasonably good correlations with experimental data under tensile loading [15–17], and has served as a means of clarifying the inﬂuence of load biaxiality on the crack growth direction. The crack growth under biaxial stress is dictated by three parameters: stress biaxiality, crack angle with respect to the applied principal stress direction a, Fig. 13, and stress intensity factor. Depending on the ﬁrst two parameters, the crack may grow in Mode I, Mode II or mixed-mode. Designating h as the angle of the tangent to the direction of initial crack growth, the criterion predicts crack growth along the radial line normal to the direction of the maximum tensile stress or along the direction parallel to the plane on which the tangential stress component rhh attains the maximum value. rhh has the following form [18]

rhh ¼ ð2prÞ1=2 cosðh=2Þ fK I cos2 ðh=2Þ ð3=2Þ K II sin hg

ð1Þ

K I ¼ frðpaÞ1=2 g fð1 þ kÞ ð1 kÞ cosð2aÞg

ð2Þ

where

K II ¼ rðpaÞ

1=2

g fð1 kÞ sinð2aÞg

ð3Þ

and

rhh =r ¼ A:½B1 fsinðh=2Þ þ sinð3h=2Þg þ B2 f3 cosðh=2Þ þ cosð3h=2Þg þ B3 sin2 h

ð4Þ

where r applied longitudinal stress (Fig. 13)

A ¼ fð1=32Þ ða=rÞg1=2

ð5Þ

B1 ¼ ðk 3Þ sin a cos a

ð6Þ

2

B2 ¼ k þ ð1 kÞ sin a

ð7Þ

B3 ¼ ð1 kÞ cos 2a

ð8Þ o

For the specimen with a transverse central notch, a = 90 , a = 0.019 m, r = 0.001 m and r = 35.7 MPa. From the above equations, h is calculated for the k values, employed in the biaxial fatigue tests, as shown in Table 3. The crack path in a biaxially stressed plate can be found from the following relationship [20]

y ¼ Cðx=DÞD where x, y is the Cartesian coordinates of a point with respect to a crack tip.

ð9Þ

1562

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564 Table 4 Calculated values of C. k

h

C

0 0.5 1 1.5

0 0 0 56o60

0 0 0 17.03/x1.576

Biaxiality Ratio 1.5

Biaxiality Ratio 0, 0.5, 1

30

80

20

Y (mm)

Y (mm)

60 40 20

0 -20 -40

10 0 -10 -20

-60 -80 -40

-20

0

X (mm)

20

40

-30 -40

-20

0

20

40

X (mm)

Fig. 14. Fatigue crack path, predicted analytically, in specimens with a transverse notch under in-phase loading.

C ¼ D:ðtan hÞ:ðD=xÞðD1Þ

ð10Þ

2

ð11Þ

D ¼ 1:145k

From the Eqs. (9)–(11), C is calculated for the values of k, employed in the biaxial tests, as shown in Table 4. From the C values of Table 4 and Eq. (9), it is clear that y becomes 0 for the all x values at k = 0, 0.5 and 1. Thus, only straight crack paths are predicted to be possible under uniaxial stressing of k = 0 and biaxial stressing of k = 0.5 and 1. On the other hand, for k = 1.5

y ﬃ 1:5x

ð12Þ

Thus, the crack path is predicted to be a straight line of slope y/x ﬃ 1.5. The actual slope, measured from the specimens fatigue-tested at k = 1.5, is 1.1–1.2. The predicted crack paths are drawn for different k values in Fig. 14. The features of the analytically predicted crack paths are similar to those observed in the biaxial fatigue tests of cruciform specimens with a transverse center notch, validating the analysis. 4.2. Observation and conﬁrmation of fatigue crack path and growth rate (Comparison of some current results with others’) Crack paths, similar to those observed in this study, were also reported by some other researchers. Kibler [19] conducted biaxial fracture tests of polymethyl methacrylate (PMMA) and an aluminum alloy, and reported: (1) at k < 1, the crack grew normally to the maximum applied load, (2) at k = 1.3, the path became S-shaped, both tips curving away from the center starter, and (3) at k = 1.8, the change in direction at the original crack tip was abrupt and pronounced. Leevers et al. [20] performed biaxial fracture tests of center-cracked PMMA plates, and observed: (1) under uniaxial loading, cracks had a strong tendency to follow the specimen center-line normal to the applied load, (2) as the transverse stress was increased, the crack path tended to deviate from this line, showing marked path instability, and (3) the path curved away from the original direction towards that normal to the maximum load for k > 1. Truchon [21] observed in the test of E36-Z steel: (1) the crack path and growth rate of Mode I fatigue crack were not modiﬁed by the stress parallel to the crack for k < 1, (2) the crack took an S-shaped path under IP loading for k > 1. Pook [22] conducted a fatigue test of a Waspaloy specimen with an initial transverse crack under in-phase loading of k = 2 and observed an S-shape fatigue crack path. His result is similar to that in this study of aluminum alloy 1100-H14 with k = 1.5. Shlvannikov [23] developed computational and experimental approaches for geometrical modeling of fatigue crack path for the inclined through the thickness central crack. The crack paths illustrated mixed-mode growth behavior of an inclined crack in specimens of aluminum alloys under k = 0.5 loading. The other researchers also investigated the variation of fatigue crack growth rate with k. Smith [3] found the fatigue crack growth rate decreasing with k becoming more positive. On the other hand, Liu and Dittmer [12] and Truchon et al. [21] claimed that k had no effect on fatigue crack growth. Leevers et al. [7] showed that there was a small difference in the effect of k for PMMA and the ductile PVC. Kitagawa et al. [4] observed: (1) the fatigue crack growth rate decreased with increase of the load amplitude parallel to a crack when the phase difference between the loads of both axes was zero, and (2) the fatigue crack growth rate increased with a change of phase difference from 0 to p for equi-biaxial load conditions. Ogura [11] reported that the fatigue crack growth rate was accelerated by OP stress while it was little affected by IP stress. McDiarmid [24] showed that an OP stress produced a shorter fatigue life than an equal IP stress.

1563

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

5. Summary 5.1. Transverse notch in 1100-H14 aluminum alloy specimen In-phase loading – Crack path is transverse for k 6 1 but deviated for k > 1. – Features of analytical crack path are similar to those of experimental one, depending on k. – Greater k induces lower crack growth rate and longer fatigue life at a given r. – Greater r and kr shorten fatigue life at a given k. Out-of-phase loading – Crack path is transverse for k 6 0.5–1.5. – Fatigue life is changed little under r 6 42.9 MPa but appreciably shortened at r = 51.5 MPa by increasing k. – Greater r and kr shorten fatigue life at a given k – Fatigue crack growth rate and life are changed little by k at a given r. – Fatigue crack growth is faster and fatigue life is shorter under OP loading than under IP loading. 5.2. Inclined notch in 7075-T651 aluminum alloy specimen IP loading k

1.5

1

0.5-0

Crack path Fatigue life

Vertical Short

Diagonal Longest

Transverse Short

Transverse

Transverse

OP loading Crack path Fatigue life

Vertical Increasing mostly with decreasing k

Greater r and kr shorten fatigue life at a given k under IP and OP loading. OP loading shortens fatigue life more with greater k or kr. 6. Conclusions Non-singular stress parallel to a crack affects the biaxial fatigue behavior of aluminum alloys, 1100-14H and 7075-T651. The path of crack, growing (or extending) from a transverse notch, can be predicted analytically and conﬁrmed experimentally.

References [1] Hilton PD. Plastic intensity factors for cracked plates subjected to biaxial loading. Int J Fract 1973;9:149–56. [2] Adams NJI. Some comments on the effect of biaxial stress on fatigue crack growth and fracture. Engng Fract Mech 1973;5:983–91. [3] Smith EW, Pascoe KJ. The behaviour of fatigue cracks subject to applied biaxial stress: a review of experimental evidence. Fat Engng Mater Struct 1983;6:201–24. [4] Kitagawa H, Yuuki R, Tohgo K. A fracture mechanics approach to high-cycle fatigue crack growth under in-plane biaxial loads. Fat Engng Mater Struct 1979;2:195–206. [5] Ang DD, Williams ML. Combined stresses in an orthotropic plate having a ﬁnite crack. Trans ASME J Appl Mech 1961;28:372–8. [6] Hopper CD, Miller KJ. Fatigue crack propagation in biaxial stress ﬁelds. J Strain Anal 1977;12:23–8. [7] Leevers PS, Radon JC, Culver LE. Crack growth in plastic panels under biaxial stress. Polymer 1976;17:627–32. [8] Christensen RH, Harmon MB. Limitation of fatigue-crack research in the design of ﬂight vehicle structures. In: Fatigue crack propagation, ASTM, STP 415; 1967. p. 5–23. [9] Joshi SR, Shewchuck J. Fatigue crack propagation in a biaxial-stress ﬁeld. Exp Mech 1970;10:529–33. [10] Habetlnek J. Fatigue strength and fatigue failure at complicated states of stress. Engng Fract Mech 1976;8:677–89. [11] Ogura K, Ohji J, Ohkubo Y. Fatigue crack growth under biaxial loading. Int J Fract 1974;10:609–10. [12] Liu AF, Dittmer DF. Effect of multiaxial loading on crack growth. Technical Summary, Final Report, September 1976–September 1978 AFFDL-TR-78175, vol. 1, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio 45433; December 1978. [13] Cotterell B. Notes on the paths and stability of cracks. Int J Fract Mech 1966;2:526–33. [14] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. Trans ASME, J Basic Engng 1963;85D:519–27. [15] Williams JG, Ewing PD. Fracture under complex stress – the angled crack problem. J Fract Mech 1972;8:441–6. [16] Finnie I, Saith A. A note on the angled crack problem and the directional stability of cracks. J Fract Mech 1973;9:484–6. [17] Jones DL, Eftis J. Fracture and fatigue characterization of aircraft structural materials under biaxial loading, Final Scientiﬁc Report, October 76–May 80, AFOSR-TR-81-0856, Air Force Ofﬁce of Scientiﬁc Research, Boling Air Force Base, Washington, DC 20032: George Washington University, Washington, DC 20052. [18] Williams ML. On the stress distribution at the base of a stationary crack. Trans ASME, J Appl Mech 1957;24:109–14. [19] Kibler JJ, Roberts R. The effect of biaxial stresses on fatigue and fracture. Trans ASME, J Engng Indust 1970;92:727–34. [20] Leevers PS, Radon JC, Culver LE. Fracture trajectories in a biaxially stressed plate. J Mech Phys Solids 1976;24:381–95.

1564

E.U. Lee, R.E. Taylor / Engineering Fracture Mechanics 78 (2011) 1555–1564

[21] Truchon M, Amestoy M, Dang-Van K. Experimental study of fatigue crack growth under biaxial loading, advances in fracture research. (Fracture 81), In: Reprints of the 5th International Conference of Fracture (ICFS), Cannes, France, 29 March–2 April 1981; 1981. p.1841–9. [22] Pook LP. Five decades of crack path research. Engng Fract Mech 2010;77:1619–30. [23] Shlyannikov VN. Fatigue crack paths for inclined cracks and surface ﬂaws under biaxial loading. Engng Fract Mech 2010;77:1772–80. [24] McDiarmid DL. Fatigue under out-of-phase biaxial stresses of different frequencies. In: Multiaxial fatigue, ASTM STP 853; 1985. p. 606–21.