Unified material parameters based on full compatibility for low-cycle fatigue characterisation of as-cast and austempered ductile irons

International Journal of Fatigue 68 (2014) 111–122

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Unified material parameters based on full compatibility for low-cycle fatigue characterisation of as-cast and austempered ductile irons B. Atzori, G. Meneghetti ⇑, M. Ricotta Department of Industrial Engineering, University of Padova, Via Venezia 1, 35131 Padova, Italy

a r t i c l e

i n f o

Article history: Received 23 December 2013 Received in revised form 20 May 2014 Accepted 29 May 2014 Available online 11 June 2014 Keywords: Low-cycle fatigue Compatibility equations Plastic strain energy Manson–Coffin curves Ductile irons

a b s t r a c t In the present paper a non-conventional procedure is proposed to determine the material parameters relevant to the low-cycle fatigue curves of as-cast and austempered ductile irons. Such a procedure is based on a set of so-called ‘full compatibility’ conditions known from the literature, and assumes the plastic strain energy as a fatigue damage parameter. Analytical coherence is ensured when deriving the material parameters. By using the full compatibility conditions, unified fatigue scatter bands are proposed to interpret the fatigue behaviour of the analysed ductile irons, which may be useful both to estimate the fatigue curves by means of a reduced number of laboratory fatigue tests and for standardisation activities. Finally, some simple correlations between static and fatigue properties of the ductile irons tested in the present paper are proposed. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Ductile irons (DIs) are engineering materials for structural applications and are suitable to manufacture mechanical components of complex geometry. Austempering heat treatments improve their mechanical properties and can be adjusted to obtain, as an example, high fatigue performances or high wear resistance. As a consequence, austempered ductile irons (ADIs) are promising materials, which may substitute cast or forged steels for demanding structural applications such as crank shafts, connecting rods and drive trains for wind turbines. DIs have microstructural features like pores and graphite nodules, which introduce discontinuities in the matrix. Several investigations are available from the literature concerning the influence of the microstructure (matrix type as well as size and shape of graphite) on static properties [1] and on high-cycle fatigue strength of smooth and notched samples [2–11]. Concerning lowcycle fatigue behaviour, these microstructural features may induce bimodular mechanical behaviour in push–pull tests due to premature crack initiation [12–14], which makes the elastic modulus in tension lower than observed in the compression part of the strain cycle; however, sometimes the stiffness reduction during the tensile part of the strain cycle has been observed only at the final stage of the fatigue test [15–18]. The results of our previous

⇑ Corresponding author. Tel.: +39 049 8276751; fax: +39 049 8276785. E-mail address: [email protected] (G. Meneghetti). http://dx.doi.org/10.1016/j.ijfatigue.2014.05.012 0142-1123/Ó 2014 Elsevier Ltd. All rights reserved.

experimental investigations support the experimental findings that a symmetrical stress–strain response of the material exists during most part of the low-cycle fatigue tests [19]. In a previous paper push–pull strain-controlled fatigue tests were carried out on ferritic, pearlitic, isothermed and austempered ductile irons covering a range of tensile strengths ru from about 400 MPa to 1300 MPa [19]. In the present paper, the approach proposed in the past by Feltner and Morrow [20], Morrow [21], Halford [22] and Ellyin [23] has been applied to analyse the fatigue behaviour of the different ductile irons by using the plastic strain hysteresis energy as a fatigue damage index. As a result, the material parameters appearing in the plastic strain energy-based curves as well as in the classic Manson–Coffin curves could be derived using a set of so-called ‘full compatibility’ expressions, which ensure analytical coherence among the parameters appearing in fatigue life equations and material constitutive laws. Full compatibility includes and extends the classical compatibility conditions, which involve only stress/strain quantities and not energy parameters as well. Classical compatibility was previously applied [19] in order to derive the strain-life, stress-life and cyclic stress–strain curve of the materials re-analysed here. Making use of the full compatibility conditions, unified fatigue scatter bands are proposed, which may be used both to estimate the fatigue properties of DIs with a reduced number of laboratory specimens and also for standardisation activities. Moreover, on the basis of the present fatigue data, some simple correlations between static and fatigue properties are proposed.

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Nomenclature af a b c E E0 K0 n0 Nf Wf W 0f DW

fatigue toughness exponent energy per cycle exponent fatigue strength exponent fatigue ductility exponent elastic modulus measured from a static tensile test dynamic elastic modulus cyclic strength coefficient cyclic strain hardening exponent number of strain cycles to failure (half the number of reversals) fatigue toughness (total plastic strain energy in a unit volume of material to fatigue fracture) fatigue toughness coefficient plastic strain energy in a unit volume of material per cycle (area of the hysteresis loop)

0

DW energy per cycle coefficient DW – Nf energy per cycle – life relation De cyclic total strain range Dr cyclic stress range ea cyclic total strain amplitude (half the range De) eap cyclic plastic strain amplitude (half the range Dep) eae cyclic elastic strain amplitude (half the range Dee) e0f fatigue ductility coefficient ru tensile strength rp02 0.2% offset yield strength ra cyclic stress amplitude (half the range Dr) r0f fatigue strength coefficient

2. Theoretical background Table 1 Static mechanical properties and Brinell hardness of the tested materials.

2.1. Basic equations Morrow [21] presented a number of compatibility conditions existing between the material constants appearing in the strainlife curve and the cyclic stress–strain curve. Moreover, he was able to show additional compatibility expressions between the strainlife equation and the plastic strain energy-life equations. While the first group of compatibility equations is often applied to determine the material constants from fatigue test results [24], the second one is rarely applied in practical fatigue design. Let us refer to constant amplitude push–pull strain-controlled fatigue tests conducted on plain specimens made of a metallic material. For the sake of simplicity, we will refer to Masing materials [25], while different approaches are available for non-Masing materials (see for example Ref. [26]). During a strain-controlled fatigue test with a given applied strain range De, the stress–strain behaviour of the material usually stabilises (in practise, when such a circumstance does not occur, the stress–strain behaviour at half

Material

a b c

Microstructure (unetched)

Material

Mean nodule diameter (lm)

ru rp02 Nod E (mm2) (GPa) (MPa) (MPa)

A HB (%)

DI 400a DI 600a DI 700a IDIb ADI 800c ADI 1050c (1st series) ADI 1050c (2nd series) ADI 1200c

45 30 35 40 35 35

220 310 244 220 244 244

160 165 161 170 170 163

440 722 805 758 858 1110

19 10 8 10 15 13

35

244

163

1160 831

12 350

35

244

148

1330 1046

7

According to [31]. According to ZANARDI STD 101: 2007. According to [32].

Microstructure (etched)

DI-400

100 μm

100 μm

ADI 1050 (2nd series) 100 μm

Fig. 1. Typical microstructure of an as-cast and a heat treated ductile iron.

100 μm

305 426 487 455 551 794

150 220 244 240 270 330

370

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fatigue life is assumed as characteristic for that applied strain range). Concerning the strain-life equation, the Basquin, Manson and Coffin equations [27–29] for the plastic Dep and the elastic Dee = Dr/E0 components of the strain are considered applicable:

The applied strain range De is considered as the sum of the elastic and the plastic contributions (De = Dee + Dep):

Dep ¼ e0f  ð2Nf Þc 2

ð1Þ

The tips of the stabilised (or half-life) hysteresis loops measured at different strain amplitudes are interpolated by the cyclic stress–strain equation:

Dr ¼ r0f  ð2Nf Þb 2

ð2Þ

σ [MPa]

500

DI 400

De r0f ¼ 0  ð2Nf Þb þ e0f  ð2Nf Þc 2 E

e ¼ eae þ eap ¼

ra E

0

 þ

ra K

 10 n

ð4Þ

0

σ [MPa]

600

DI 600

ð3Þ

400

400

300 200

200

100 0 -0.007

ε [m/m] C

-0.0035 -100 0

0.0035

0.007

ε [m/m]

0 -0.008

-0.004

0

0.004

0.008

-200

-200 -300

-400

-400

-600

-500 σ [MPa]

700

DI 700

500

525

300

350

100 -0.01

175

ε [m/m]

-0.005 -100 0

0.005

0.01

-0.01

-0.005 -175 0

0.005

0.01

-350

-500

-525

-700

-700 σ [MPa]

800

750

400

500 250

ε [m/m]

0 -0.005 -200 0

0.005

0.01

-0.01

-0.005 -250 0 -500

-600

-750

-800

-1000

σ [MPa]

ADI 1050 (2 series) 1000

ε [m/m]

0

-400

nd

σ [MPa]

ADI 1050 (1st series) 1000

600

200

-0.01

ε [m/m]

0

-300

ADI 800

σ [MPa]

700

IDI

750

0.01

σ [MPa]

1200

ADI 1200

0.005

800

500 400

250

ε [m/m]

-0.01

-0.005

-250

0

0.005

0.01

ε [m/m]

0

0 -0.008

-0.004

0

0.004

0.008

-400

-500 -800

-750 -1000

-1200

Fig. 2. Typical stabilised or half-life hysteresis loops measured during the strain-controlled fatigue tests.

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Concerning the fatigue life curves based on the plastic strain hysteresis energy, the following energy-life equations are assumed applicable [21]:

DW ¼ DW 0  ð2Nf Þa

ð5Þ

W f ¼ W 0f  ð2Nf Þaf

ð6Þ

where DW is the stabilised (or half-life) plastic strain energy per cycle (i.e. area of the stabilised or half-life hysteresis loop) and Wf is the total plastic strain energy at specimen’s fracture, also referred to as fatigue toughness [21]. The apexes indicate the quantities referred to one reversal 2Nf = 1 (the same meaning is attributed to the material constants r0f and e0f appearing in Eqs. (1) and (2), respectively). Compatibility equations given by Morrow [21] can be summarised as follows: 1) Compatibility among parameters in strain-life (1) and cyclic stress–strain (4) equations

n0 ¼

b c

ð7Þ

K0 ¼

r0f 0 ðe0f Þn

ð8Þ

1) Compatibility among parameters in strain-life (3) and energy-life (5) and (6) equations

a¼bþc

 Austenitization between 800 and 900 °C for about 2 h.  Quenching in a salt bath (NaNO3 and NaNO2) between 300 and 400 °C for about 1.5 h.  Cooling at room temperature. Isothermed ductile irons contains significantly smaller amounts of alloying elements as compared to austempered ductile irons [19,33]. Thanks to their peculiar microstructure consisting of ferrite-pearlite interconnected, isothermed ductile irons have mechanical properties which are intermediate between those of conventional ferritic–pearlitic and austempered ausferritic grades. IDI materials may substitute low grade ADIs in the case of high volume productions or castings characterised by heavy wall thicknesses [33]. IDI and ADI specimens were machined by turning after heat treatment. Samples made of each material came from the same pouring batch and, when applicable, from the same heat treatment batch. In particular, DI 700 and ADI specimens came from a single pouring batch. The final microstructures of the DI 400 and of the ADI 1050 material are reported as an example in Fig. 1. Details about specimens’ geometry, static tension tests and strain-controlled fatigue tests are reported elsewhere [19]. Here

ð9Þ 1.E+5

DW 0 ¼ 4  a  r0f  e0f

ð10Þ

af ¼ 1 þ a

ð11Þ

DW 0 ¼ 2  W 0f

ð12Þ

The first set of expressions ((7) and (8)) are the compatibility conditions, whereas both sets (Eqs. (7)–(12)) define the full compatibility conditions. For a Masing material, the branches of the hysteresis loops are geometrically similar to the cyclic stress–strain curve (Eq. (4)) and are magnified by a factor equal to two. Under this hypothesis, Halford [22] derived the following analytical expression to calculate the area of the hysteresis loop:

DW ¼

and fatigue tests were prepared from Lynchburg raw specimens having diameter 25 mm and length 200 mm. Isothermed and austempered materials were obtained by keeping the heat treatment parameters within the following range:

1  n0  Dr  Dep 1 þ n0

ð13Þ

1.E+4

1.E+3

1.E+2

1.E+1 1.E+2

1.E+3

1.E+4

1.E+5

1.E+6

2Nf, number of reversals to failure Fig. 3. Fatigue toughness against number of reversals to failure.

1n0 1þn0

being a ¼ the material constant appearing in Eq. (10). Different expressions of a may be applicable, depending on the assumed shape of the hysteresis loop [30]. The first group of compatibility Eqs. (7) and (8) can be easily derived by eliminating the number of cycles to failure, Nf, from Eqs. (1) and (2). Compatibility Eqs. (9) and (10) can be derived by using expressions (1) and (2) into Eq. (13), and Eqs. (11) and (12) descend from the assumption that the following expression holds true:

W f ¼ DW  Nf

ð14Þ

and by substitution of expression (5) into (14). 3. Materials and low-cycle fatigue test results A number of ferritic, pearlitic (DI grade), isothermed (IDI grade) and austempered (ADI grade) ductile irons were analysed in a previous paper [19]. A selection of those materials and the relevant mechanical properties are reported in Table 1. All samples for static

Fig. 4. Unified scatter band of the fatigue toughness against fatigue life.

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1.E+5

1.E+5

Wf [MJ/m3]

1.E+4

1.E+4

1.E+3

1.E+3

1.E+2

Wf = 101 ⋅ (2 N f )0.358 T10/90=2.19

1.E+2

Wf [MJ/m3]

Wf = 167 ⋅ (2 N f )0.358 T10/90=2.19

DI 400

1.E+1 1.E+5

1.E+1 1.E+5

Wf [MJ/m3]

1.E+4

1.E+4

1.E+3

1.E+3

0.358 1.E+2 Wf = 160 ⋅ (2 N f ) T10/90=2.19

1.E+2 DI 700

1.E+1 1.E+5 1.E+4

1.E+4

1.E+3

1.E+3

1.E+2

Wf = 185 ⋅ (2 N f )0.358 T10/90=2.19

W f [MJ/m3]

Wf = 214 ⋅ (2 N f )0.358 T 10/90=2.19 IDI

1.E+1 1.E+5

Wf [MJ/m3]

1.E+2

Wf [MJ/m3]

Wf = 158 ⋅ (2 N f )0.358 T10/90=2.19

ADI 1050 (1st series)

ADI 800

1.E+1 1.E+5

1.E+1 1.E+5

Wf [MJ/m3]

DI 600

Wf [MJ/m3]

1.E+4

1.E+4

1.E+3

1.E+3

0.358 1.E+2 Wf = 135 ⋅ (2 N f ) T10/90=2.19

0.358 1.E+2 Wf = 123 ⋅ (2 N f ) T10/90=2.19

1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 2Nf, number of reversals to failure

1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 2Nf, number of reversals to failure

ADI 1050 (2nd series)

ADI 1200

Fig. 5. Unified scatter band in terms of fatigue toughness fitted to each series of experimental results. Slope af = 0.358 and scatter index T10/90 = 2.19 from Fig. 4.

we recall that the fatigue tests were performed using round specimens having diameter 5 mm. A MTS FPF10 servo-hydraulic testing machine equipped with a 100 kN load cell, a TRIO Sistemi digital controller and a MTS extensometer with a gauge length of 25 mm were adopted. Fatigue tests were conducted in a range of fatigue cycles between 50 and 105 cycles by imposing a sinusoidal waveform characterised by a nominal strain ratio Re (ratio between the minimum and the maximum applied strain) equal to 1. The load test frequency was kept between 0.25 and 3 Hz. The adopted failure criterion was the specimen’s separation.

During strain-controlled fatigue tests, stress–strain hysteresis loops were recorded. Concerning the DI and IDI grades, it was noted that the measured stress amplitude slightly increased during the tests according to a hardening behaviour. Concerning the ADI grades, it was noticed that, if the applied strain amplitude was greater than 0.5%, then the material rapidly strain hardens at the beginning of the fatigue tests, and later a slight softening occurs. Conversely, if the strain level is lower than 0.5%, the material response is stable during the fatigue test. Stabilised or half-life hysteresis loops are reported in Fig. 2.

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Table 2 Unified cyclic material parameters of the tested materials taking into account full compatibility. Material

E0 (GPa)

n0 Eq. (7)

W 0f

DI 400 DI 600 DI 700 IDI ADI 800 ADI 1050 (1st series) ADI 1050 (2nd series) ADI 1200

161

0.136

101 167 160 214 185 158 135 123

h

MJ m3

i

af

DW0

0.358

202 334 320 428 370 316 270 246

h

MJ m3 cycle

Fig. 6. Stabilised or half-life plastic strain energy per cycle against fatigue life.

Fig. 7. Unified scatter band of the stabilised or half-life stress amplitude against fatigue life.

The materials analysed here showed almost identical slopes of the decreasing and increasing linear parts of the stabilised (or half-life) hysteresis loop, from which the elastic component was evaluated. The plastic strain was then calculated as difference between the total strain range and the elastic strain range:

Depl ¼ De  Deel

ð15Þ

4. Data reduction 4.1. Principal diagrams and material parameters To apply the full compatibility approach, it seems convenient to start by analysing the fatigue toughness against life equation

i

Eq. (12)

a Eq. (11)

K0 (MPa) Eq. (8)

r0f (MPa)

b

e0f

c Eq. (9)

0.642

966 1202 1268 1284 1537 2039 2145 2565

700 903 937 989 1133 1444 1473 1733

0.077

0.094 0.122 0.108 0.147 0.106 0.079 0.063 0.056

0.565

because material stabilisation during fatigue tests must not be necessarily assumed. Fig. 3 reports the fatigue toughness against the number of reversals to failure. The resulting slope of the mean curve is af = 0.358, which is in good agreement with the value 0.33 found by Halford [22]. The scatter index referred to the 10% and 90% survival probability curves results 2.88, and reduces to 2.19 as soon as the intrinsic statistical scatter is analysed according to Fig. 4, where each test series is normalised by using W f calculated from a regression analysis in correspondence to 2N f equal to 2000 fatigue cycles. Fig. 5 shows the fair agreement of the unified scatter band reported in Fig. 4 with each test series. Full compatible fatigue toughness coefficients are calculated from this figure and are reported in Table 2. Fig. 6 reports the stabilised or half-life plastic strain energy per cycle against the fatigue life. Interestingly, the plastic strain energy per cycle is seen to be independent of the material static strength, at least as a first approximation [34]. In fact the scatter index calculated with 10% and 90% survival probability curves is 2.90 which is not much larger than the intrinsic scatter index equal to 2.19, as will be shown in Fig. 10. Fig. 7 reports the unified scatter band of the stabilised or half-life stress amplitude versus the fatigue life and shows a fatigue strength exponent b = 0.077 to use in Eqs. (2) and (3). Such a value is in agreement with b = 0.09 proposed by Muralidharan and Manson [35] for steels, aluminium and titanium alloys. Data reported in Fig. 7 have been normalised using the same approach adopted in Fig. 4. Fig. 8 shows the degree of accuracy of the unified scatter band when fitted to the experimental results of each test series. There is some level of approximation concerning the DI 400 material where the experimental results claim for a shallower slope. Full compatible fatigue strength coefficients are calculated from this figure and are reported in Table 2. Finally, Fig. 9 shows the stabilised or half-life cyclic stress amplitude as a function of the applied elastic strain amplitude and highlights that the dynamic elastic modulus can be assumed equal to E0 = 161,000 MPa for all materials with a reduced scatter of the experimental results. The unified values of af, b and E0 determined from Figs. 4, 7 and 9, respectively, are also reported in Table 2.

4.2. Material parameters derived from principal diagrams Fig. 10 shows the scatter band fitted on the normalised values of the plastic strain energy per cycle. Again, DW* is the value relevant to 2N f ¼ 2000 fatigue cycles. It should be noted that the unified scatter band is derived from the principal one, see Fig. 4, being the slope in Fig. 10 equal to a = 0.3581 = 0.642 due to compatibility condition (11). The scatter index has been set equal to 2.19 according to the result shown in Fig. 4. Fig. 11 shows a comparison between the unified scatter band reported in Fig. 10 and the experimental results. It should be noted that the scatter bands in Fig. 11 are not fitted on the experimental results, but they are simply superimposed since both slope and intercept are determined from

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1000

1000

σa [MPa]

σ a = 700 ⋅ (2 N f )−0.077 T10/90=1.16

1000

σ a = 903 ⋅ (2 N f )−0.077 T10/90=1.16

DI 400

100

1000

σa [MPa]

σa [MPa]

σ a = 989 ⋅ (2 N f )−0.077 T10/90=1.16 DI 700

100

IDI

100 1000

σa [MPa]

σa [MPa]

σa = 1444 ⋅ (2 N f )−0.077 T10/90=1.16

σ a = 1133 ⋅ (2 N f )−0.077 T10/90=1.16

ADI 1050 (1st series)

ADI 800

100

1000

DI 600

100

σ a = 937 ⋅ (2 N f )−0.077 T10/90=1.16

1000

σa [MPa]

100

2000

σa [MPa]

σa [MPa]

1000

σ a = 1473 ⋅ (2 N f )−0.077 T10/90=1.16

σ a = 1733 ⋅ (2 N f )−0.077

T10/90=1.16

nd

ADI 1050 (2 series)

100 1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 2Nf, number of reversals to failure

ADI 1200

100 1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 2Nf, number of reversals to failure

Fig. 8. Unified scatter band in terms of stabilised cyclic stress amplitude fitted to each series of experimental results. Slope b = 0.077 and scatter index T10/90 = 1.16 from Fig. 7.

Fig. 9. Cyclic stress against elastic strain amplitudes.

Fig. 10. Unified scatter band of the plastic strain energy per cycle as a function of the fatigue life. Slope a = 0.3581 = 0.642 from Eq. (11), scatter index T10/90 from Fig. 4.

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compatibility expressions (11) and (12), respectively. Agreement between experimental results and superimposed scatter bands is doubtless satisfactory. The energy per cycle coefficients calculated from Eq. (12) and the energy per cycle exponent are also reported in Table 2. After separating the plastic strain component from the total applied strain (Eq. (15)), the unified scatter band reported in Fig. 12 is determined using normalised values. Again it should be noted that the slope c = 0.565 shown in the figure is obtained from compatibility condition (9), where the slopes b and a are known from Fig. 7 and Eq. (11), respectively. The value 0.565 is in excellent agreement with c = 0.56 proposed by Muralidharan and Manson [35] for steels, aluminium and titanium alloys. In an previous publication, Manson had proposed c = 0.6 [36], while Coffin suggested a value c = 0.5 [29]. The unified scatter band in Fig. 12 is fitted to the experimental results in Fig. 13, where the fair agreement between the imposed slope and the experimental

1.E+2

Fig. 12. Unified scatter band of the cyclic plastic strain amplitude against fatigue life. Slope c = 0.642 + 0.077 = 0.565 from Eq. (9).

1.E+2

ΔW [MJ/(m3 ⋅ cycle)]

1.E+1

1.E+1

1.E+0

1.E+0

1.E-1

ΔW = 202 ⋅ (2 N f )−0.642 T10/90=2.19

1.E-1 DI 400

1.E-2 1.E+2

1.E+2

ΔW [MJ/(m3 ⋅ cycle)]

1.E+1

1.E+0

1.E+0 ΔW = 320 ⋅ (2 N f )−0.642 T10/90=2.19 DI 700

1.E-2

1.E+2

1.E-1

1.E+2

ΔW [MJ/(m3 ⋅ cycle)]

1.E+1

1.E+0

1.E+0 ΔW = 370 ⋅ (2 N f )−0.642 T10/90=2.19

1.E-1 ADI 800

1.E-2

1.E+2

ΔW [MJ/(m3 ⋅ cycle)]

1.E+2 1.E+1

1.E+0

1.E+0

1.E-2

ΔW [MJ/(m3 ⋅ cycle)]

ΔW = 428 ⋅ (2 N f )−0.642 T10/90=2.19

IDI

ΔW [MJ/(m3 ⋅ cycle)]

ΔW = 316 ⋅ (2 N f )−0.642 T10/90=2.19 ADI 1050 (1st series)

1.E-2

1.E+1

1.E-1

DI 600

1.E-2

1.E+1

1.E-1

ΔW = 334 ⋅ (2 N f )−0.642 T10/90=2.19

1.E-2

1.E+1

1.E-1

ΔW [MJ/(m3 ⋅ cycle)]

ΔW = 270 ⋅ (2 N f )−0.642 T10/90=2.19

1.E-1

ADI 1050 (2nd series)

1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 2Nf, number of reversals to failure

ΔW [MJ/(m3 ⋅ cycle)]

ΔW = 246 ⋅ (2 N f )−0.642 T10/90=2.19

ADI 1200

1.E-2 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 2Nf, number of reversals to failure

Fig. 11. Unified scatter band in terms of plastic strain energy per cycle superimposed to each series of experimental results. Slope a = 0.642 from Eq. (11), intercept DW0 from Eq. (12), scatter index T10/90 = 2.19 from Fig. 10.

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results can be appreciated. Full compatible fatigue ductility coefficients e0f are calculated from this figure and reported in Table 2. The last scatter band analysed in the present paper is reported in Fig. 14, where the normalised stress amplitudes measured after stabilisation or at half-life are reported against the plastic strain amplitude. The slope n0 in Fig. 14 derives from compatibility condition (7). The unified scatter band reported in Fig. 14 is superimposed to each test series in Fig. 15, where it should be noted that both the slope (n0 ) and the intercept (K0 ) are fixed by virtue of compatibility expressions (7) and (8), respectively. Cyclic strength coefficients K0 and strain hardening exponents n0 have been reported in Table 2. The degree of accuracy of the unified scatter band when superimposed to the experimental results can be appreciated. Indeed a certain degree of approximation exists for DI 400 and DI 600 grades. In particular the slope of the scatter band is steeper than what the experimental results would suggest.

1.E-1

Fig. 14. Unified scatter band of the cyclic stress against plastic strain amplitude. Slope n0 = 0.077/(0.565) = 0.136 from Eq. (7).

1.E-1

ε ap

1.E-2

1.E-2

1.E-3

1.E-3

1.E-4

ε ap = 0.094 ⋅ (2 N f )−0.565 T10/90=2.72

1.E-4 DI 400

1.E-5

1.E-1 1.E-2

1.E-2

1.E-3

1.E-3

1.E-4

ε ap = 0.108 ⋅ (2 N f )−0.565 T10/90=2.72

1.E-4 DI 700

1.E-5

1.E-1

1.E-1

ε ap

1.E-2

1.E-3

1.E-3 ε ap = 0.106 ⋅ (2 N f )−0.565 T10/90=2.72

ADI 800

1.E-5 1.E-1

1.E-4

1.E-2

1.E-3

1.E-3

1.E-5

1.E-4

ε ap = 0.063 ⋅ (2 N f )−0.565 T10/90=2.72

nd

ADI 1050 (2 series)

1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 2Nf, number of reversals to failure

1.E-5

IDI

ε ap

ε ap = 0.079 ⋅ (2 N f )−0.565 T10/90=2.72

1.E-2

1.E-4

ε ap = 0.147 ⋅ (2 N f )−0.565 T10/90=2.72

1.E-5 1.E-1

ε ap

DI 600

ε ap

1.E-5

1.E-2

1.E-4

ε ap = 0.122 ⋅ (2 N f )−0.565 T10/90=2.72

1.E-5

1.E-1

ε ap

ε ap

ADI 1050 (1st series)

ε ap

ε ap = 0.056 ⋅ (2 N f )−0.565 T10/90=2.72

ADI 1200

1.E+1 1.E+2 1.E+3 1.E+4 1.E+5 1.E+6 1.E+7 2Nf, number of reversals to failure

Fig. 13. Unified scatter band in terms of cyclic plastic strain amplitude fitted to each series of experimental results. Slope c = 0.565 from Eq. (9) and scatter index T10/90 = 2.72 from Fig. 12.

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2000

2000

σa [MPa]

σa [MPa]

1000

1000

σ a = 1202 ⋅ ε 0ap.136

σ = 966 ⋅ ε0ap.136

Serie2 a

T10/90=1.17

100 2000

DI 400

T10/90=1.17

2000

σa [MPa]

1000

σa [MPa]

1000

σa = 1268 ⋅ ε0ap.136

σa = 1284 ⋅ ε 0ap.136

T10/90=1.17

DI 700

100 2000

T10/90=1.17

IDI

100

2000

σa [MPa]

1000

σa [MPa]

1000

σ a = 1537 ⋅ ε 0ap.136 T10/90=1.17

100 2000

DI 600

100

σ a = 2039 ⋅ ε 0ap.136 T10/90=1.17 ADI 1050 (1st series)

ADI 800

100 2000

σa [MPa]

1000

σa [MPa]

1000

σ a = 2145 ⋅ ε 0ap.136 T10/90=1.17

100 1.E-4

ADI 1050 (2nd series)

1.E-3 1.E-2 εap [m/m]

1.E-1

σ a = 2565 ⋅ ε 0ap.136 T10/90=1.17

100 1.E-4

ADI 1200

1.E-3 1.E-2 εap [m/m]

1.E-1

Fig. 15. Unified scatter band in terms of cyclic stress against plastic strain amplitudes superimposed to each series of experimental results. Slope n0 = 0.136 from Eq. (7), intercept K0 from Eq. (8) and scatter index T10/90 = 1.17 from Fig. 14.

4.3. Correlation between static and fatigue properties In this section an analysis is proposed in order to estimate the coefficients of the fatigue curves starting from the static properties found from a tensile test. Fig. 16 reports the fatigue toughness coefficient W 0f as a function of the strain hardening ratio, defined as the ratio between

the tensile and the offset yield strengths ratio [24]. The figure suggests that two linear lines can be drawn for as-cast and heat treated materials, respectively. For the same strain hardening ratio, about 50 MJ/m3 more energy is required to fracture a heat treated than an as-cast ductile iron. Fig. 17 shows the fatigue strength coefficient as a function of the tensile strength. It is interesting to note that DI 400 (fully ferritic) and DI 700 (fully pearlitic) markers

B. Atzori et al. / International Journal of Fatigue 68 (2014) 111–122

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5. Discussion

Fig. 16. Fatigue toughness coefficient as a function of strain hardening ratio.

Fig. 17. Fatigue strength coefficient

Fig. 18. Fatigue ductility coefficient

r0f as a function of tensile strength.

e0f as a function of strain hardening ratio.

fall slightly above and below the mean regression line, respectively. Finally, Fig. 18 reports the fatigue ductility coefficient as a function of the strain hardening ratio. The trend is increasing, though a certain scatter exists.

A method has been proposed which accounts for the so-called full compatibility between energy-based and stress/strain-based fatigue life equations. When the full set of compatibility expressions is applied, a choice has to be made on which material parameters shall be used as principal parameters and which ones should be derived from compatibility expressions. Having in hands the hysteresis loops measured for each specimen during the entire fatigue test, the diagram relating the total plastic strain energy to fracture (i.e. the fatigue toughness Wf) against the fatigue life is chosen as principal one. This choice seems reasonable in light of the fact that the total hysteresis energy to fracture takes into account the evolution of the material behaviour during the fatigue tests. Therefore Wf seems a convenient parameter to start the analysis, in particular when the material does not attain stabilisation during the fatigue test and consequently the half-life quantities represent only approximately the overall fatigue behaviour of the material from the beginning of the test until final fatigue fracture. Figs. 4 and 5 show that a unified scatter band in terms of slope and scatter index can be proposed to interpret the fatigue toughness-life experimental data. The second principal diagram adopted in the present paper is the stabilised (or half-life) stress amplitude against the total fatigue life. Figs. 7 and 8 show that unified values of the slope and scatter index can be proposed for all ductile irons tested in the present work. Figs. 10–15 are derived from the principal diagrams, because slopes or both slopes and intercepts are calculated from full compatibility conditions. Unified exponents of the stabilised cyclic curve as well as of the fatigue curves are summarised in Table 2 along with the relevant coefficients. For comparison purposes, Table 3 reports the material parameters calculated using the compatibility conditions Eqs. (7) and (8) according to the procedure outlined in a previous paper [19]. It should be noted that, in the latter case, each material has been treated apart from the remaining ones, leading to material constants that are all specific of the tested materials and thus may be desirable for design purposes. On the other hand, unified slopes of the fatigue curves and scatter bands proposed in the present paper are useful: (i) to analyse statistically the results of low cycle fatigue tests, which is not a conventional approach since strain-life curves are traditionally presented only in terms of mean fatigue life, while statistical scatter is not considered [37]; (ii) to speed up the determination of Manson–Coffin curves using a reduced number of specimens; in fact few fatigue data can be fitted with the unified scatter bands, where both the slope and the scatter index are fixed; (iii) to standardise fatigue curves of ductile irons subject to different heat treatments. Moreover, the strength coefficients ðW 0f ; r0f ; e0f Þ of each material can be made to depend on static properties within a certain level of approximation, as Figs. 16–18 illustrate. It should be noted that the fatigue strength of nodular cast iron can be made to depend on two material parameters, i.e. the matrix hardness and a representative value of the maximum nodule size [38]. However, the materials analysed in the present paper have the same statistical distribution of the nodule size, at least as a first approximation. In fact, all Lynchburg raw specimens were prepared using the same production process so that the mean nodule diameter and the nodule count is similar for the different materials as reported in Table 1, with the exception of the DI 600 ductile iron. Therefore, the strength coefficients W 0f , r0f and e0f are expressed solely as a function of the static mechanical properties ru and rp02 in Figs. 16–18. It is the authors opinion that the rules shown in Figs. 16–18 are valid for ductile iron materials having mean nodule diameter and nodule count within the range reported in Table 1.

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Table 3 Cyclic material parameters of the tested materials taking into account compatibility (from [19]). Material

E0 (GPa)

n0

K0 (MPa)

r0po2 (MPa)

r0f (MPa)

b

e0f

c

DI 400 DI 600 DI 700 IDI ADI 800 ADI 1050 (1st series) ADI 1050 (2nd series) ADI 1200

164 165 165 159 157 156 158 167

0.069 0.092 0.082 0.134 0.125 0.173 0.163 0.175

624 906 903 1265 1420 2598 2568 3361

406 511 542 550 653 887 933 1133

557 770 807 935 995 1789 1576 2174

0.046 0.056 0.057 0.070 0.060 0.106 0.085 0.110

0.194 0.171 0.254 0.104 0.057 0.116 0.050 0.083

0.667 0.610 0.690 0.521 0.476 0.613 0.524 0.629

6. Conclusions In this paper, a set of so-called full compatibility conditions are applied to define unified scatter bands which are useful to analyse the results of push–pull strain-controlled low-cycle fatigue tests. Eight series of as-cast and heat treated ductile irons are considered. Experimental data processing starts from the fatigue toughnesslife and cyclic stress-life equations. The slope of such curves are seen unique for all tested materials and, in particular, the fatigue strength exponent b resulted 0.077. By applying the full compatibility conditions, unified values of the fatigue ductility exponent c and of the cyclic strain hardening exponent n0 equal to 0.565 and 0.136, respectively, are calculated. Unified scatter indexes of the elastic strain-life and plastic strain-life curves are also determined and are equal to 2.16 and 2.72, respectively, when referred to the 10% and 90% survival probabilities. Finally, the relation between static properties and strength/ductility coefficients of the strainlife fatigue curves is analysed. As a result, three relations are proposed to estimate the fatigue toughness, the fatigue strength and fatigue ductility coefficients, respectively. As an outcome, the strain-life equations for a ductile iron can be estimated solely on the basis of static mechanical properties and of the unified scatter bands proposed in the present paper, provided that the mean nodule diameter and nodule count is within the range analysed here. Due to the averaging effect of a statistical re-analysis of experimental results, care should be taken when using the unified scatter bands for accurate durability evaluations of a structural component made of a specific ductile iron. Acknowledgements This work has been carried out as a part of the Italian Research Program PRIN 2009Z55NWC of the Ministry of University and Scientific Research. The Authors would like to express their gratitude for the financial support. References [1] Putatunda SK. Development of austempered ductile cast iron (ADI) with simultaneous high yield strength and fracture toughness by a novel two-step austempering process. Mater Sci Eng A 2001;31:70–80. [2] Faubert GP, Moore DJ, Rundman KB. Heavy-section ADI: fatigue properties in the as-cast and austempered condition. AFS Trans 1991;111:563–70. [3] Bartosiewicz L, Krause AR, Alberts FA, Singh I, Putatunda SK. Influence of microstructure on high-cycle fatigue behavior of austempered cast iron. Mater Charact 1993;30:221–34. [4] Lin CK, Lai PK, Shih TS. Influence of microstructure on the fatigue properties of austempered ductile irons-I. High-cycle fatigue. Int J Fatigue 1996;18: 297–307. [5] Chapetti MD. High-cycle fatigue of austempered ductile iron (ADI). Int J Fatigue 2007;29:860–8. [6] Carpinteri A, Spagnoli A, Vantatori S. Size effect in S–N curves: a fractal approach to finite-life fatigue strength. Int J Fatigue 2009;31:927–33. [7] Atzori B, Bonollo F, Meneghetti G. Notch fatigue and fracture mechanics of austempered ductile irons. Key Eng Mater 2011;457:181–6. [8] G. Meneghetti, S. Masaggia, Estimation of the fatigue limit of components made of Austempered Ductile Iron weakened by V-shaped notches. In: Proceedings of the 70th World Foundry Congress WFC 2012, Monterrey, Mexico.

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