Crack initiation and growth path under multiaxial fatigue loading in structural steels

International Journal of Fatigue 31 (2009) 1660–1668

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Crack initiation and growth path under multiaxial fatigue loading in structural steels L. Reis *, B. Li, M. de Freitas Department of Mechanical Engineering, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

a r t i c l e

i n f o

Article history: Received 12 September 2008 Received in revised form 30 December 2008 Accepted 14 January 2009 Available online 27 January 2009 Keywords: Multiaxial fatigue Fatigue life Cyclic plasticity Crack initiation Crack path Additional hardening

a b s t r a c t In real engineering components and structures many accidental failures occur due to unexpected or additional loadings, such as additional bending or torsion. There are many factors influencing the fatigue crack paths, such as the material type (microstructure), structural geometry and loading path. It is widely believed that fatigue crack nucleation and early crack growth are caused by cyclic plasticity. This paper studies the effects of multiaxial loading paths on the cyclic deformation behaviour, crack initiation and crack path. Three types of structural steels are studied: Ck45, medium carbon steel, 42CrMo4, low alloy steel and the AISI 303 stainless steel. Four biaxial loading paths were applied in the tests to observe the effects of multiaxial loading paths on the additional hardening, fatigue crack initiation and crack propagation orientation. Fractographic analyses of the plane orientations of crack initiation and propagation were carried out by optical microscope and SEM approaches. It is shown that these materials have different crack orientations under the same loading path, due to their different cyclic plasticity behaviour and different sensitivity to non-proportional loading. Theoretical predictions of the damage plane were conducted using the critical plane approaches, either based on stress analysis or strain analysis (Findley, Smith–Watson–Topper, Fatemi–Socie, Wang–Brown–Miller, etc). Comparisons of the predicted crack orientation based on the critical plane approaches with the experimental observations for the wide range of loading paths and the three structural materials are shown and discussed. Results show the applicability of the critical plane approaches to predict the fatigue life and crack initial orientation in structural steels. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Most engineering components are subjected to multiaxial states of stress and strain due to the presence of stress raisers such as notches or holes. Therefore, growing research efforts have been paid to study the cyclic deformation behaviour and fatigue life prediction of engineering materials and components under multiaxial loading conditions [1]. For engineering design, crack initiation life is commonly defined as crack nucleation and small crack growth to NDT detectable crack size. Different mechanisms and controlling parameters dominate the fatigue life stages. A dominant fraction of total fatigue life may be spent in the growth of small cracks (less than approximately 1–2 mm). The problem of the nucleation and growth of small cracks has been conventionally treated in the context of so-called fatigue crack initiation approaches, which are phenomenological stress-life and strain-life equations. A new potentiality of assessing the fatigue strength and service life of structural component arises from small crack fracture mechanics. In recent years, substantial effort has been directed to model the small crack growth behaviour [2–4], since an * Corresponding author. Fax: +351 218417915. E-mail addresses: [email protected] (L. Reis), [email protected] (M. de Freitas). 0142-1123/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2009.01.013

improved understanding of the ability to predict small crack growth behaviour under multiaxial stress conditions would be very useful for fatigue design evaluations. In real engineering structures, there are many factors influencing the fatigue crack initiation and crack paths, such as the material type (microstructure), structural geometry and loading path. It is widely believed that fatigue crack nucleation and early crack growth are caused by cyclic plasticity in structural steels. Forsyth [5] has designated two stages of crack growth, stage I where cracks grow along the planes of maximum shear stress and stage II where cracks grow on the plane normal to the direction of the maximum principal stress. Kanazawa et al [6] studied the endurance and the direction of crack growth in 1Cr–Mo–V steel under out-of-phase axial/torsional loading condition. Significant influences of out-of-phase loading conditions were shown. Crack paths orientations have received increasing attention in the fatigue research field [7–11]. The objective of this paper is to present the experimental and theoretical studies about the effects of multiaxial loading paths on the cyclic deformation behaviour, crack initiation and the early crack growth on three types of structural steels. In particular, it is focused on the effects of multiaxial, proportional and non-proportional, loading paths, on the additional hardening and the influence of the microstructure on fatigue crack initiation and early crack propagation orientation. It is shown that these materials have

L. Reis et al. / International Journal of Fatigue 31 (2009) 1660–1668

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Nomenclature

sa

fatigue life (failure cycle) material parameter for a given fatigue life Nf material parameter for a given fatigue life Nf shear stress range shear strain range normal stress

Nf k(Nf) k(Nf) Ds Dc

r

SEM NDT LCF HCF

different crack orientations under the same loading path, due to their different cyclic plasticity behaviour and different sensitivity to non-proportional loading, which also influences their fatigue life and early crack growth orientation. Results also show the applicability of the critical plane approaches to predict the fatigue life and crack initial orientation in structural steels. 2. Material data, specimens form and test procedures Three structural steels are studied in this work: a plain carbon Ck45 steel in the normalized condition, which microstructure is formed by ferrite and perlite and the AISI 303 stainless steel, which has a microstructure mainly austenitic and a high strength quenched and tempered at 525 °C steel, 42CrMo4, which microstructure is bainitic. To characterize the uniaxial cyclic stress– strain behaviour of the materials, tension–compression low cycle fatigue tests were carried. The cyclic properties were obtained by fitting the test results according to the standard procedures. The

Table 1 Chemical composition of Ck45, AISI 303 and 42CrMo4 steels in (wt%).

CK45 AISI 303 42CrMo4

C

Si

Mn

P

S

Cr

Ni

Mo

Cu

0.45 0.12 0.39

0.21 1.00 0.17

0.56 2.00 0.77

0.018 0.060 0.025

0.024 0.250 0.020

0.10 18.0 1.10

0.05 9.00 0.30

0.01 – 0.16

0.10 – 0.21

Table 2 Monotonic and uniaxial cyclic mechanical properties of the studied materials.

Tensile strength Yield strength Young’s modulus Elongation Cyclic yield strength Strength coefficient Strain hardening exponent Fatigue strength coefficient Fatigue strength exponent Fatigue ductility coefficient Fatigue ductility exponent

a

ru (MPa) r0.2% (MPa) E (GPa) A (%) r00:2% (MPa) ´ (MPa) K ´ n r0f r´f (MPa) b

e0f c

Ck45

AISI 303

42CrMo4

660 410 206 23 340 1206 0.20 948 0.102 0.17 0.44

625 330 178 58 310 2450 0.35 534 0.07 0.052 0.292

1100 980 206 16 640 1420 0.12 1154 0.061 0.18 0.53

amplitude of the alternating shear stress scanning electron microscope non destructive testing low cycle fatigue high cycle fatigue

chemical compositions and monotonic and cyclic mechanical properties are shown in Tables 1 and 2, respectively. Despite the differences in the monotonic mechanical properties, these three steels can be considered as presenting a ductile behaviour. The geometry and dimensions of the specimens used in the testing program are shown in Fig. 1. The specimens were polished before testing. In order to characterize and analyze the effects of the loading paths on the additional hardening and consequently on the cyclic stress–strain behaviour of the studied materials, a series of LCF and HCF tests, under proportional and non-proportional axial/torsional paths, as shown in Fig. 2, thick black line, were carried out using a biaxial servo-hydraulic machine (8800 Instron) and a data acquisition system, see Fig. 3. In Fig. 2 the outer circle means the same von Mises equivalent strain/stress, which is equal for all the cases by von Mises criterion. Under these four strain loading paths, different cyclic stress responses were observed and different models are applied considering the respective damage parameter, i.e., taking into account the different values of the normal and shear components of stress/strain. The loading path 3 is sequential axial/torsion loadings. There are two ways to consider the damage of sequential axial/torsion loadings: one way is to consider the axial loading and torsional loading separately, and then to sum the damages. Another way, as in this paper, is to consider one sequential axial/torsion loading block as one cycle, and then consider the damage from the whole cycle. All low cycle fatigue tests were carried out using a biaxial extensometer and fully reversed of sinusoidal and trapezoidal waves (R = 1, strain controlled), at a range of frequency between 0.01–0.05 Hz and performed at room temperature. This range of frequency allows an equivalent strain rate of 3.5  104 s1. Fatigue failure life is defined as the number of cycles at which a 10% drop from the maximum value occurs in either the tensile or shear stress range. To study the effects of the multiaxial loading paths on the fatigue crack orientation paths, the same series of loading paths, as shown in Fig. 2, were applied in the experiments, but now in load control. Test conditions were at a frequency of 4–6 Hz at room temperature in laboratory air. Tests ended up when the specimens were completely broken. Then, fractographic analysis of the macroscopic plane of crack initiation and early crack growth were carried out, using an optical microscope at a magnification between

b

Fig. 1. Specimens geometry for: (a) uniaxial/biaxial LCF tests and (b) biaxial HCF tests (all dimensions in mm).

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γ

γ

3

ε

ε

Case 1

γ

3

Case 2

γ

3

ε

Case 3

3

ε

Case 4

Fig. 2. Biaxial axial/torsional fatigue loading strain/stress paths.

10 and 100 times. Some of the fractured specimens were also analyzed in SEM. The measurement of the crack initiation plane orientation was carried out as follows: firstly, the site of crack initiation was identified, as indicated by a white arrow, see Fig. 4a); then, the specimen was analyzed in a 3D measurement device and the angle between the crack initiation plane and the longitudinal axis was accurately measured, as shown on Fig. 4b). Experimental measured plane angles for all specimens and materials, respectively, are presented in Section 4. This study was carried out considering the HCF life regime of 105–106 cycles. 3. Multiaxial fatigue damage models From the microscopic scale, the mechanism of crack initiation in ductile metal crystals is the alternating dislocation movements along a slip plane in a grain [9]. This slip is introduced by an alter-

nating shear stress in the slip plane. If the amplitude of the alternating shear stress sa exceeds a critical value, a crack will initiate and then will grow out of the grain and becomes a macroscopic crack. Therefore, it is generally assumed that crack initiation in ductile materials is mainly controlled by the amplitude of alternating shear stress sa. A common feature of many multiaxial fatigue criteria is that they are expressed as a general form and include both shear stress amplitude sa and normal stress r during a loading cycle:

sa þ kðNf Þr ¼ kðNf Þ

ð1Þ

Multiaxial fatigue models differ in the interpretation of how shear stress amplitude sa and normal stress r in Eq. (1) are defined, [12,13]. For example, the Findely [14], McDiarmid [15] and DangVan [16] criteria use forms of the range of maximum shear stress, while the Sines [17] and Crossland [18] criteria make use of the octahedral shear stress range. Mean stresses have been incorporated into the models by using either the hydrostatic stress (such as Sines, Crossland and Dang-Van) or the normal stress on a plane (such as McDiarmid). The stress-invariant based models take into account the shear stress over all planes, because the damage parameter is defined as the amplitude of the second stress-invariant, which represents the root-mean-square of the shear stresses over all planes. In contrast to the critical plane models [14,19–25], the integral models [26,27] define the fatigue damage parameter as the integral value of shear stresses over all planes instead the amplitude of the shear

Fig. 3. Biaxial testing machine: (a) LCF experimental setup and (b) HCF experimental setup.

Fig. 4. Measurement of the crack initiation plane orientation of proportional loading path, case 1, to CK45 material: (a) identification of local of crack initiation and (b) measure angle of the crack initiation plane.

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stress only on the critical plane. Since the critical plane models are capable to predict both fatigue life and crack propagation orientation, they were applied in this study. 3.1. Theoretical analysis with critical plane models For the biaxial loading cases 1–4, shown in Fig. 2, both the fatigue life and the potential crack plane orientation are analyzed by various critical plane models and energy-based critical plane models, such as the Findley, the Brown–Miller, the Fatemi–Socie, the Smith–Watson–Topper and the Liu’s criteria. For the three structural materials, multiaxial fatigue models gave different damage parameters by introducing the material-dependent parameter in the respective damage parameter formulations. 3.1.1. Findley model Based on physical observations of the orientation of initial fatigue cracks in steel and aluminum, Findley [14] discussed the influence of normal stress acting on the maximum shear stress plane. A critical plane model was introduced, which predicts that the fatigue crack plane is the plane of orientation h which maximize the Findley damage parameter, Eq. (2):

  Ds þ krn ¼ f ðhÞ 2

ð2Þ

where Ds/2 is the shear stress amplitude on a plane h, rn is the maximum normal stress on that plane and k is a material parameter. For the case of finite long-life fatigue it comes:

  Ds þ krn ¼ sf ðNf Þb 2

ð3Þ

where sf is computed from the torsional fatigue strength coefficient, s0f . 3.1.2. Brown and Miller model Analogous to the shear and normal stress proposed by Findley [14], Brown and Miller [19] proposed that both the shear and normal strain on the plane of maximum shear must be considered. Cyclic shear strains will help to nucleate cracks, and the normal strain will assist in their growth. Later, Kandil, Brown and Miller [20] proposed a simplified formulation of the theory for case A cracks: 1

Dc^ ¼ ðDcamax þ SDean Þa

ð4Þ

^ is the equivalent shear strain range and S is a materialwhere Dc dependent parameter that represents the influence of the normal strain on material crack growth and it is determined by correlating axial and torsion data. More recently, Wang and Brown [21] added a mean stress term to the formulation and assuming a = 1, the equivalent shear strain amplitude was formulated as:

Dc^ Dcmax ¼ þ SDen 2 2

ð5Þ

where Dcmax is taken as the maximum shear strain range and Den is the normal strain range on the plane experiencing the maximum shear strain range Dcmax . To compute fatigue life Eq. (6) can be considered:

r0f Dcmax þ SDen ¼ A ð2N f Þb þ Be0f ð2N f Þc 2 E

ð6Þ

where A = 1.3 + 0.7S and B = 1.5 + 0.5S. 3.1.3. Fatemi–Socie model The Fatemi–Socie (F–S) model [22] is widely applied for shear damage model, which predicts that the critical plane is the plane of orientation h with the maximum F–S damage parameter:

  Dc rn;max 1þk ¼ f ðhÞ 2 ry

ð7Þ

where Dc/2 is the maximum shear strain amplitude on a plane h, rn,max is the maximum normal stress on that plane, ry is the material monotonic yield strength; k is a material constant, which can be found by fitting fatigue data from simple uniaxial tests to fatigue data from simple torsion tests, k = 0.4, 1.0 and 0.2 for Ck45, 42CrMo4 and AISI 303, respectively. To compute fatigue life Eq. (8) can be considered:

  s0f Dc rn;max 1þk ¼ ð2N f Þbc þ c0f ð2N f Þcc 2 ry G

ð8Þ

where G is the shear modulus, s0f is the shear fatigue strength coefficient, c0f is the shear fatigue ductility coefficient, and bc and cc are shear fatigue strength and shear fatigue ductility exponents, respectively. These properties can be obtained from torsion fatigue test, or they can be estimated from uniaxial strain-life properties [1]. As an example, Fig. 5 shows the variations of Fatemi–Socie damage parameter during one loading cycle in 3D dimension, for the proportional loading path 1 and for the non-proportional loading path 2 for the AISI 303 stainless steel. From Fig. 5 is possible identify two positive peaks in case 1 and one positive peak in case 2. 3.1.4. Smith–Watson–Topper model The S–W–T tensile damage model, proposed by Smith, Watson and Topper [23], predicts that the fatigue crack plane is the plane of orientation h with maximum normal stress (the maximum principal stress):

rn;max

De1 ¼ f ðhÞ 2

ð9Þ

where De1/2 is the maximum principal normal strain amplitude and rn,max is the maximum normal stress on the De1 plane. To compute fatigue life Eq. (10) can be considered:

rn;max

De1 r02 f ¼ ð2N f Þ2b þ e0f r0f ð2Nf Þbþc 2 E

ð10Þ

3.1.5. Liu’s virtual strain-energy model The Liu’s virtual strain energy (VSE) model [24] is an energybased critical plane model. This model considers two possible failure modes: a mode for tensile failure, DWI, and a mode for shear failure, DWII. Failure is expected to occur on the material plane h having the maximum VSE quantity. DWI is computed by firstly identifying the plane on which the axial work is maximized and then adding the respective shear work on the same plane, Eq. (11):

DW I ¼ ðDrn Den Þmax þ ðDsDcÞ

ð11Þ

h

Fig. 6 presents the maximum variations of the Liu WI parameter on different planes for the four loading cases, to the 42CrMo4 material. To compute fatigue life Eq. (12) can be considered: 2

DW I ¼ 4r0f e0f ð2Nf Þbþc þ

4r0f E

ð2Nf Þ2b

ð12Þ

Similarly, DWII is computed by firstly identifying the plane on which the shear work is maximized and then adding the axial work on that same plane, Eq. (13) and fatigue life can be achieved considering Eq. (14):

DW II ¼ ðDrn Den Þ þ ðDsDcÞmax

ð13Þ

h

DW II ¼ 4s0f c0f ð2Nf Þbc þcc þ

4s02 f G

ð2Nf Þ2bc

ð14Þ

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Fig. 5. The variations of Fatemi–Socie damage parameter during one loading cycle in 3D: (a) for the proportional loading path 1 and (b) for the non-proportional loading path 2.

where Ds and Dc are the shear stress range and shear strain range, respectively, Drn and Den are the normal stress range and normal strain range, respectively. For each loading case and for each model the critical plane angle h was identified and achieved. Results are summarized and presented in the next section. 4. Results and discussion 4.1. Cyclic stress–strain behaviour and fatigue life under proportional and non-proportional loading Proportional cyclic tests were conducted in three directions in p the e  c/ 3 plot: at 0°, at 45° and at 90°. Non-proportional cyclic tests were conducted with the circular; cross and square paths, respectively (see Fig. 2). Fig. 7a, 8a and 9a show one example of the cyclic stress evolution during controlled strain low cycle fatigue loading, case 3, and Fig. 7b, 8b and 9b show the cyclic stress–strain responses at the stabilized cycles for the four different loading paths for Ck45, AISI 303 and 42CrMo4, respectively. Results show that the AISI 303 steel is a very sensitive material to non-proportional loading paths, as can be observed through the cross path presented in Fig. 8a and the overall results presented in Fig. 8b. The AISI 303 stainless steel has a face-centered cubic (FCC) structure and has low stacking fault energy with widely spaced dislocations [1,28]. During non-proportional loading, the planes of maximum shear stress rotate, thus initiating plastic deformation along several different slip systems. Stresses are continuously increasing during the first 10–15 cycles until stabilization is

Liu parameter W I [MPa]

7.0

Case 1

Case 2

Case 3

Case 4

6.0 5.0 4.0 3.0 2.0 1.0 0.0 -90

-75

-60

-45

-30

-15

0

15

30

45

60

75

90

Plane angle [º] Fig. 6. Maximum variations of the Liu WI parameter on different planes for the four loading cases to 42CrMo4 material.

reached. Against this behaviour the Ck45 and the 42CrMo4 materials present a relatively low sensitivity to non-proportional loading, as can be seen by one example of cyclic stress evolution presented in Fig. 7a and 9a). Other aspect achieved by the results is that a very significant difference of fatigue behaviour was observed under proportional or non-proportional loading conditions. Non-proportional loading conditions, i.e. for non-proportional straining, a rotation of the principal strain axes leads to an additional cyclic strain hardening as compared with that obtained under stable conditions for proportional stressing. In the latter case, dislocations glide on a fixed set of slip planes, whereas for continuously rotating principal axes, many slip systems are activated. Therefore an additional factor should be introduced to take into account the dependence of the loading history. To quantify the level of non-proportionality of the loading path, several ways and some mathematical descriptions have been proposed in the literature [29–31]. Another important parameter is the so-called additional hardening coefficient. This coefficient is a constant of a given material, which is defined as the ratio of equivalent stress under 90° out-ofphase loading and the equivalent stress under proportional loading at high plastic strains in the flat portion of the stress–strain curve. The degree of additional hardening observed in the 90° out-ofphase loading case is dependent on the ease with which multiple cross-slip systems develop in a given material [1,28,29,32–34]. This coefficient reflects the maximum level of additional hardening that might occur for a given material. The additional hardening coefficient is highly dependent on the microstructure and development of slip systems in a material. In this paper, the additional hardening coefficient is evaluated and compared for the three materials at room temperature, as shown in Table 3. Figs. 7b, 8b and 9b show the cyclic stress–strain responses in terms of equivalent stress amplitude versus equivalent strain amplitude at the stabilized cycles for Ck45, AISI 303 and 42CrMo4, respectively. As can be seen, at the same equivalent strain amplitude, the non-proportional loading paths caused much more hardening than the proportional paths for all materials studied, i.e. the additional hardening is also very dependent on the intensity of the loading, i.e. as higher is the loading (Deeq. = 0.60%) more influence has in the effect of hardening, comparing with Deeq. = 0.25%. It is also possible to conclude that the cross and the circular loading cases are the most influent concerning the additional hardening in both situations, i.e when Deeq. = 0.25% and Deeq. = 0.60%. The values presented in Table 3 confirm that AISI 303 is the most sensitive material to non-proportional loading paths, i.e. exhibits the higher additional hardening effect. Through a close

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L. Reis et al. / International Journal of Fatigue 31 (2009) 1660–1668

b 750

750

500

600

250 0 -750

-500

-250 0 -250

250

500

750

-500

Equiv. Stress (MPa)

Sqrt(3)*Shear stress (MPa)

a

450 300

Monotonic Tens-Compr. Torsion Proportional Square Cross Circle

150 0

-750

Normal Stress (MPa)

0.0

0.2

0.4

0.6

0.8

1.0

Equivalent Strain (%)

Fig. 7. Material Ck45: (a) stress evolution during controlled strain, loading case 3 and (b) cyclic stress–strain responses at the stabilized cycles for the six different loading paths.

a Sqrt(3)*Shear stress (MPa)

750 500 250 -750

-500

0 -250-250 0

250

500

750

-500

Equiv. Stress (MPa)

b

1000

-750

900

600 450 300 150 0

-1000

Monotonic Tens-Compr. Torsion Proportional Square Cross Circle

750

Normal Stress (MPa)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Strain (%)

Fig. 8. Material AISI 303: (a) stress evolution during controlled strain, loading case 3 and (b) cyclic stress–strain responses at the stabilized cycles for the six different loading paths.

a

b

1050

1050 700 350 -1200

-800

-400

0 -350

1200

0

400

800

1200

-700 -1050 -1400

Normal Stress (MPa)

Equiv. Stress (MPa)

Sqrt(3)*Shear stress (MPa)

1400

900 750 600 Monotonic Tens-Compr. Torsion Proportional Square Cross Circle

450 300 150 0 0.0

0.2

0.4

0.6

0.8

1.0

1. 2

Equivalent Strain (%)

Fig. 9. Material 42CrMo4: (a) stress evolution during controlled strain, loading case 3 and (b) cyclic stress–strain responses at the stabilized cycles for the six different loading paths.

comparison with the cyclic behaviour of AISI 304 stainless steel [4– 6,17,18], a similar increase in hardening is observed. Relative to the mechanism of overhardening of stainless steel, there are some explanations: the slip mechanism of the dislocation is believed to play an important role; the degree of non-proportional hardening increases as the material stacking fault energy decreases. Another factor is due to the stress-induced phase transformation as observed in our tests for AISI 303 stainless steel. Therefore, it seems that the above two mechanisms are present in the studied material. In terms of strength to fatigue life, Table 4 presents the number of cycles obtained for the three materials tested and a comparison is also made between the predicted life cycles by the Fatemi–Socie model and experimental results. As can be observed, under the

same equivalent strain amplitude but with different strain paths, from proportional (case 1) to non-proportional biaxial loading paths (cases 2, 3 and 4), significant differences were observed on the fatigue life, where the non-proportional biaxial loading paths showed reduced number of fatigue cycles, i.e. fatigue life is strongly dependent on the loading path for the same equivalent strain range.

Table 3 Values of additional hardening constant obtained for three materials studied. Additional hardening constant

Ck45

AISI 303

42CrMo4

0.3

0.9

0.18

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Table 4 Values of fatigue life obtained for the studied materials. Materials

Strain path case

Axial strain De2 (%)

Shear strain Dc2 (%)

Stabilized axial stress Dr2 (MPa)

Stabilized shear stress Ds2 (MPa)

Predicted life_cycles Fatemi–Socie

Experimental life_cycles Nf

Ck45

1 2 3 4

0.424 0.60 0.60 0.424

0.734 1.039 1.039 0.734

300 434 420 418

210 356 340 316

1272 374 384 385

818 277 220 349

AISI 303

1 2 3 4

0.424 0.60 0.60 0.424

0.734 1.039 1.039 0.734

270 630 550 530

184 490 420 401

1514 183 212 220

1000* 255 280 265

42CrMo4

1 2 3 4

0.424 0.60 0.60 0.424

0.734 1.039 1.039 0.734

480 750 720 690

419 564 620 510

448 110 115 120

350 134 166 232

*

Not fail. Test stopped.

Considering the lifetime reduction, for instance between cases 1 and 2, there are very significant reductions for the three materials. For the same equivalent von Mises stress/strain states, it was obtained a reduction of 70%, 75% and 87% for Ck45, 42CrMo4 and AISI 303, respectively. The correlation between the prediction and the experimental results are very good. 4.2. Crack initiation and orientation of the early crack growth under biaxial loading Through careful observations on the fracture surfaces of the specimens, the early crack plane orientations of stage I propagation, stage II propagation and final fracture were identified. It was found that the stage II propagation and final fracture showed very irregular crack orientations for the specimens tested. In this paper, only the orientations of stage I propagation were studied. Crack orientations of the stage I propagation under various loading paths were characterized and measured by microscopy. Fig. 10 presents an example of the fracture surface of each material under the same loading path, i.e case 2. A white arrow indicates the initiation local of the crack. Fig. 11 presents some micrographics of the microstructure in the local of crack initiation under the same

loading path, case 2, for the three materials. As can be seen, to the same solicitation there is a different response from the microstructure, the final morphology is quite different between each other. Table 5 presents the measured values for early crack initiation for all loading path studied and these values are compared with the predicted results by the theoretical models. Fractographic analyses of the fracture surface and early crack orientation are helpful for identifying the effects of the non-proportional multiaxial loading. Analysing Table 5 it is possible to conclude that the loading paths have a strong influence on the crack orientations, i.e. from case 1 to case 4, the early crack orientations vary significantly. Since both 42CrMo4 and Ck45 steels are ductile materials, similar crack orientations were observed for both materials under the same loading paths, except for square loading path where the crack orientation was observed to be 15° for 42CrMo4 and 15° for Ck45. Despite AISI 303 also be a ductile material, the obtained results to this material are quite different from previous ones. The theoretical predictions of the critical plane by the multiaxial fatigue models are compared with the orientations of early crack growths measured in experiments under various loading paths. The multiaxial fatigue models can be classified as shear-based

Fig. 10. Fractographic analysis of the fatigue failure plane orientation under the loading path 2 (90° shift) for: (a) Ck45; (b) AISI 303 and (c) 42CrMo4 materials, respectively.

Fig. 11. Fractographic analysis of the microstructure in the local of crack initiation under the loading path 2 for: (a) Ck45; (b) AISI 303 and (c) 42CrMo4 materials, respectively.

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L. Reis et al. / International Journal of Fatigue 31 (2009) 1660–1668 Table 5 Comparison of the measured crack plane with predictions for the studied materials. Case 1

Life Nf

Case 2

Life Nf

Case 3

Life Nf

Case 4

Life Nf

949000 974000 965000

0° 0° 24°

978000 924000 981000

5° 0° 0°

916000 964000 933000

15° 5° 32°

954000 914000 939000

Crack plane measured

42CrMo4 Ck45 AISI 303

16° 20° 25°

Findley

42CrMo4 Ck45 AISI 303

16°/65° 18°/66° 15°/63°

0° 0° 0°

0° 0° 0°

±21° ±21° ±22°

Brown–Miller

42CrMo4 Ck45 AISI 303

14°/63° 18°/66° 16°/65°

0° 0° 0°

0° 0° 0°

±21° ±21° ±21°

Fatemi–Socie

42CrMo4 Ck45 AISI 303

14°/63° 18°/66° 18°/68°

0° 0° 0°

0° 0° 0°

±21° ±21° ±21°

S–W–T

42CrMo4 Ck45 AISI 303

25° 25° 25°

0° 0° 0°

0° 0° 0°

±25° ±25° ±25°

Liu I

42CrMo4 Ck45 AISI 303

25° 25° 25°

0° 0° 0°

0° 0° 0°

±25° ±25° ±25°

Liu II

42CrMo4 Ck45 AISI 303

21°/69° 21°/69° 21°/69°

0°/±90° 0°/±90° 0°/±90°

0°/±90° 0°/±90° 0°/±90°

±21°/±69° ±21°/±69° ±21°/±69°

models (Findley, Wang–Brown, Fatemi–Socie and Liu II) and tensilebased models (S–W–T and Liu I). As shown in the Table 5, the shearbased models gave better predictions than the tensile-based models, which were expected for the ductile materials, for which the dominant failure mechanism is shear crack nucleation and growth. From the shear-based models, Findley’s criterion gives the best prediction, where the maximum deviation is 6°, in the square loading path. It is interesting to note that some times a large number of nucleated cracks were observed under a microscope in the crack initiation stage, similar phenomenon’s had also been observed by Ohkawa et al. [35], but only one main crack is formed in the early propagation, as observed in our tests. Therefore, only this main crack orientation was considered in this study. The approximate size of fatigue crack is about 1–2 mm. The microstructural features of the material do influence the early crack growth, which contributes to differences between the predicted crack orientation and the test results. On the interpretations of the Findley and Fatemi–Socie parameters for non-proportional loading paths, some recent publications (see Bonnen and Topper [36]) suggest that the normal stress on a plane and the shear stress on the plane do not necessarily need to occur at exactly the same point in time. In other words, a normal tensile stress on a plane will still enhance the shear growth on that plane even if they do not both occur at the same point in time. In this paper, the common interpretation of the Findley parameter and F–S parameter for non-proportional loading paths is applied and the normal stress and the shear stress considered occur at the same time point [1].

5. Conclusions The fatigue cyclic plasticity behaviour and the early crack growth orientation of three structural materials under various proportional and non-proportional multiaxial loading paths were both experimentally and analytically studied. Based on this study some conclusions can be drawn: – The local cyclic stress/strain states are influenced by the multiaxial loading paths, due to the interactions between the normal stress and shear stress during cyclic plastic deformation.

– At the same equivalent strain amplitude, the non-proportional loading paths caused much more hardening than the proportional paths, which has a great influence on the decreasing of the fatigue life. – The microstructure is an important key and has a great influence on the additional hardening. The additional hardening effect is dependent on the loading path and also the intensity of the loading. The calculated values of additional hardening coefficient have a very good correlation with the values presented in literature. – The studied structural materials have different early crack orientations under the same loading path. – The shear-based multiaxial models (Findley, Wang–Brown, Fatemi–Socie and Liu II) give good predictions of the orientation of the crack initiation plane for the material 42CrMo4 steel, but less accurate predictions were obtained for the AISI 303 stainless steel due to its big additional cyclic hardening effect. This observation appears to show that the critical plane models are applicable for structural steels, but may give big errors for metals with large cyclic plasticity such as the stainless steel. – To evaluate fatigue damage under complex multiaxial loading conditions, ductility-dependent damage mechanism need to be considered, appropriate fatigue models should be selected and applied. Acknowledgements The authors gratefully acknowledge financial support for this work from FCT – Fundação para Ciência e Tecnologia, through the Project POCTI/EME/59577/2004. References [1] Socie D, Marquis G. Multiaxial fatigue. Warrendale (PA): Society of Automotive Engineers; 2000. [2] Park J, Nelson D, Rostami A. Small crack growth in combined bending-torsion fatigue of A533B steel. Fatigue Fract Eng Mater Struct 2001;24:179–91. [3] McDowell DL, Bennett VP. Small crack growth in combined bending-torsion fatigue of A533B steel. Fatigue Fract Eng Mater Struct 1996;19:821–37. [4] Reddy SC, Fatemi A. In: Mitchell MR, Landgraf RW, editors. ‘‘Small crack growth in multiaxial fatigue”, advances in fatigue lifetime predictive techniques. ASTM

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L. Reis et al. / International Journal of Fatigue 31 (2009) 1660–1668

STP 1122. Philadelphia: American Society for Testing and Materials. p. 276–98. [5] Forsyth PJE. Proceedings of the crack propagation symposium, Vol. 1. Cranfield: The College of Aeronautics; 1961. p. 76–94. [6] Kanazawa K, Miller KJ, Brown MW. J Eng Mat Techol, Trans ASME 1977;99(3): 222–8. [7] Carpinteri A, editor. Handbook of fatigue crack propagation in metallic structures. Amsterdam: Elsevier Science BV; 1994. [8] Pook LP. Introduction to fatigue crack paths in metals. In: Proceedings of the international conference on fatigue crack path, 18–20 September, Parma, Italy; 2003. [9] Radaj D, Sonsino CM. Fatigue assessment of welded joints by local approaches. Cambridge: Abington Publishing; 1999. [10] Miller KJ, McDowell DL. In: Miller KJ, McDowell DL, editors. Mixed-mode crack behavior. ASTM STP 1359. West Conshohocken (PA): ASTM; 1999. p. 7–19. [11] Reis L, Li B, Leite M, Freitas M. Fatigue Fract Eng Mater Struct 2005;28: 445–54. [12] Zenner H. Multiaxial fatigue – methods, hypotheses and applications – an overview. In: Proceedings of the 7th international conference on biaxial and multiaxial fatigue and fracture, Berlin, Germany; 2004. p. 3–16. [13] Sonsino CM, Zenner H, Yousefi-Hashtyani F, Kuppers M. Present limitations in the assessment of components under multiaxial service loading. In: Proceedings of the 7th international conference on biaxial and multiaxial fatigue and fracture, Berlin, Germany; 2004. p. 17–25. [14] Findley WN. A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending. J Eng Ind 1959:301–6. [15] McDiarmid DL. A general criterion for high cycle multiaxial fatigue failure. Fatigue Fract Eng Mater Struct 1991;14:429–53. [16] Dang-Van K. In: McDowell DL, Ellis R, editors. Advances in multiaxial fatigue. ASTM STP 1191. Philadelphia: ASTM; 1993. p. 120–30. [17] Sines G. Behavior of metals under complex static and alternating stresses. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw Hill; 1959. p. 145–69. [18] Crossland B. In: Proceedings of the international conference on fatigue of metals; 1956. [19] Brown MW, Miller KJ. A theory for fatigue under multiaxial stress–strain conditions. In: Proceedings of the institute of mechanical engineers, vol. 187; 1973. pp. 745–56. [20] Kandil FA, Brown MW, Miller KJ. Biaxial low-cycle fatigue fracture of 316 stainless steel at elevated temperatures, book 280. London: The Metals Society; 1982. p. 203–10.

[21] Wang CH, Brown MW. Life prediction techniques for variable amplitude multiaxial fatigue – part 1: theories. J Eng Mater Technol, Trans ASME 1996;118(3):367–70. [22] Fatemi A, Socie D. A critical plane approach to multiaxial fatigue damage including out-of-phase loading. Fatigue Fract Eng Mater Struct 1988;11(3): 149–65. [23] Smith RN, Watson P, Topper TH. A stress–strain function for the fatigue of metal. J Mater 1970;5(4):767–78. [24] Liu KC. A method based on virtual strain energy parameters for multiaxial fatigue life prediction. In: McDowell DL, Ellis R, editors. Advances in Multiaxial Fatigue, ASTM STP 1191; 1993. p. 67–84. [25] Papadopoulos IV. Critical plane approaches in high-cycle fatigue: on the definition of the amplitude and mean value of the shear stress acting on the critical plane. Fatigue Fract Eng Mater Struct 1998;21:269–85. [26] Zenner H, Simburger A, Liu J. On the fatigue limit of ductile metals under complex multiaxial loading. Int J Fatigue 2000;22:137–45. [27] Kueppers M, Sonsino CM. Assessment of the fatigue behaviour of welded aluminium joints under multiaxial spectrum loading by a critical plane approach. Int J Fatigue 2006;28(5–6):540–6. [28] McDowell DL, Stahl D, Stock S, Antolovich S. Biaxial path dependence of deformation substructure of type 304 stainless steel. Metall trans A – Phys Metall Mater Sci 1988;19(5):1277–93. [29] Itoh T, Sakane M, Ohnami M, Socie DF. Nonproportional low cycle fatigue criterion for type 304 stainless steel. ASME J Eng Mater Technol 1995;117: 285–92. [30] Doong SH, Socie DF. Constitutive modelling of metals under nonproportional cyclic loading. J Eng Mater Technol 1991;113:23–30. [31] de Freitas M, Li B, Santos JLT. Multiaxial fatigue and deformation: testing and prediction, ASTM STP 1387; 2000. [32] Ellyin F. Fatigue damage, crack growth and life prediction. London (UK): Chapman & Hall; 1997. [33] Chen X, Xu S, Huang D. A critical plane-strain energy density criterion for multiaxial low cycle fatigue life under non-proportional loading. Fatigue Fract Eng Mater Struct 1999;22:679–86. [34] Sonsino CM. Influence of load and deformation-controlled multiaxial tests on fatigue life to crack initiation. Int J Fatigue 2001;23(2):159–67. [35] Ohkawa I, Takahashi H, Moriwaki M, Misumi M. A study of fatigue crack growth under out-of-phase combined loadings. Fatigue Fract Eng Mater Struct 1997;20(6):929–40. [36] Bonnen JJF, Topper TH. Effect of bending overloads on torsional fatigue in normalised 1045 steel. Int J Fatigue 1999;21(1):23–33.