Nd i

.K

Engin

tion,

form

line 2

igue

mon

redic

nts f

tigue

D3p 0 c amplitude versus cyclic life for aluminum alloys investi-

International Journal of FatiguE-mail address: afatemi@eng.utoledo.edu (A. Fatemi).1. Introduction

The strain-based approach to fatigue is widely used for

different materials at present. Strain-life fatigue curves,

which are also often called low-cycle fatigue curves, are

plotted on loglog scales and total strain amplitude is

resolved into elastic and plastic strain components based on

data from the steady-state hysteresis loops [1]. Basquin [2]

observed that for steel and copper materials the stress-life

data could be linearized on loglog scale. The line can be

represented by

Ds

2Z sa Z s

0f2Nfb (1)

where Ds/2 is true stress amplitude, 2Nf is reversals tofailure, s0f is fatigue strength coefficient, and b is fatiguestrength exponent. Coffin and Manson found the plastic

strain-life data could also be linearized on loglog scale.

This line can be expressed as

where D3p/2 is plastic strain amplitude, 30f is fatigue ductility

coefficient, and c is fatigue ductility exponent. The total

strain amplitude can then be considered as the summation of

elastic and plastic amplitudes and the resulting strain-life

curve can be expressed as:

D3

2Z 3a Z

D3e2

CD3p2

Zs0fE

2Nfb C30f2Nfc (3)

The life at which elastic and plastic components of strain

are equal is called the transition fatigue life (2Nt). For lives

shorter than 2Nt the deformation is mainly plastic, whereas

for lives longer than 2Nt the deformation is mainly elastic.

Endo and Morrow [3] observed that for 2024-T4 and

7075-T6 aluminum alloys they investigated the usual linear

loglog relations between fatigue life and elastic and plastic

strains do not provide adequate correlations of the test

results. They recommended using actual fatigue data plots

for these materials rather than simple power functions.

Sanders et al. [4] showed that plots of plastic strainApplication of bi-linear loglog S

data of aluminum alloys an

A. Fatemia,*, A. Plaseieda, A

aDepartment of Mechanical, Industrial, and ManufacturingbGeneral Motors Corpora

Received 13 May 2004; received in revised

Available on

Abstract

Bi-linear loglog model is applied to stress amplitude versus fat

model provides a much better representation of the data than the com

stressstrain, stress-life, and strain-life curves are discussed. Life p

also compared and discussed. Estimations of the bi-linear fit consta

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Fatigue of aluminum alloys; Fitting of Al alloys fatigue data; Famodel to strain-controlled fatigue

ts effect on life predictions

. Khosrovanehb, D. Tannerb

eering, The University of Toledo, Toledo, OH 43606, USA

Warren, MI 48090, USA

23 January 2005; accepted 12 March 2005

3 May 2005

life data of 14 aluminum alloys. It is shown that the bi-linear SN

ly used linear model for Al alloys. The effects of bi-linear model on

tions of aluminum alloys based on linear and bi-linear models are

rom the linear fit constants are then presented.

properties of Al alloys; Life prediction of Al alloys

e 27 (2005) 10401050

www.elsevier.com/locate/ijfatiguelevel is alloy dependent and below this critical level there is

a departure from single slope behavior. Therefore, in the

lower plastic strain region, the CoffinManson relationship

does not obey the single slope behavior. They related this

deviation from single slope behavior of a CoffinManson0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijfatigue.2005.03.003

* Corresponding author. Tel./fax: C1 419 530 8213.2Z 3f2Nf (2) gated reflect linearity of the CoffinManson relationship

down to a critical level of plastic strain. This plastic strain

Nomenclature

b fatigue strength exponent in the linear model

b1, b2 fatigue strength exponent in the bi-linear model

in region I, in region II

c fatigue ductility exponent

E modulus of elasticity

Kt stress concentration factor

K 0 cyclic strength coefficientn 0 cyclic strain hardening exponent2Nf reversals to failure

2Ns, 2Nt separation, transition fatigue life

Sa nominal stress amplitude

Sf fatigue limit

Sy, S0y monotonic, cyclic yield strength

Su ultimate tensile strength

3aZD3/2 total strain amplitude30f fatigue ductility coefficientD3e/2, D3p/2 elastic, plastic strain amplitudesaZDs/2 true stress amplitudesmax maximum stress

s0f fatigue strength coefficient in the linear models0f1, s

0f2 fatigue strength coefficient in the bi-linear model

in region I, in region II

Dss/2 stress amplitude at separation fatigue life in thebi-linear model

A. Fatemi et al. / International Journal of Fatigue 27 (2005) 10401050 1041plot to the relative inability of the microstructure to develop

homogeneous slip during low plastic strain cycling. Wong

[5] investigated several aluminum alloys and reported that

the stress amplitude with the logarithm of total life did not

form a linear relationship. For both the plastic strain and

stress amplitudes, the data trends show a distinct downward

concavity. Later, Stephens and Koh [6] reported and

discussed bi-linearity of stress amplitude versus fatigue

life for three types of A356-T6 cast aluminum alloys. They

attributed the cause of this bi-linearity to the somewhat flat

nature of the cyclic stressstrain curve in the inelastic region

of these Al alloys. Bi-linear loglog behavior of elastic

strain versus fatigue life has also occurred in other

aluminum alloys [79]. Ignoring the non-linearity can result

in large data scatter leading to significant inaccuracies in life

predictions.

In this work, based on fatigue data obtained for 14

aluminum alloys, bi-linear loglog stress-life behavior of

these alloys is presented and the effects of this bi-linearity

on cyclic stressstrain, stress-life, and strain-life curves are

discussed. The effect of bi-linearity on life prediction ofTable 1

Summary of monotonic tensile and strain-controlled fatigue properties of alumin

Alloy Process descrip-

tion

E (GPa) Sy (MPa) Su (MPa) S

6063 Extruded profile 73.4 239 263 2

A356-T6 Casting 78.1 232 303 2

6260 Extruded profile 70.5 220 239 2

6063 Extruded tube 71.9 161 192 1

5754-NG Sheet 65.7 107 253 2

6082 Extruded bar 64.0 290 2

AlMg4.5Mn Sheet 71.5 298 363 3

5456-H311 Bar 69.1 235 400 3

7475-T761 Sheet 70.0 414 475 4

7075-T6 Sheet 72.2 512 572 3

7075-T651 Sheet 70.0 5

7075-T7351 Sheet 71.0 382 462 3

2014-T6 Bar 69.1 463 511 4

2024-T3 Sheet 70.3 345 490 4

a Sf is fatigue limit calculated from s0f (10

9)b.aluminum alloys is then discussed, and estimations of bi-

linear fit constants from linear fit constants are presented.

Table 1 shows the summary of monotonic tensile and strain-

controlled fatigue properties of the 14 aluminum alloys used

in this study. Data for the first six alloys were obtained as a

part of this study, whereas data for other alloys were taken

from [10].

Monotonic tension and constant amplitude fully reversed

fatigue tests for the alloys tested in this study were performed

using test methods specified by ASTM Standards E8 and

E606, respectively [11]. Flat plate specimens with square

cross section and uniform gage section length, as shown in

Fig. 1 were used. The relatively short gage section length was

chosen to prevent buckling during compression in fatigue

tests. A closed-loop servo-hydraulic 50 kN axial load frame

in conjunction with a digital servo-controller and hydraulic-

wedge grips was used to conduct the tests. Significant effort

was put forth to align the load train and minimize bending.

Total strain was controlled using an extensometer with a gage

length of 6 mm and rated as class B1, according to ASTM

classification. In order to protect the specimen surface fromum alloys

0y (MPa) s

0f (MPa) 3

0f b c Sf

a (MPa)

54 556 0.74 K0.107 K0.830 60.1

91 666 0.09 K0.117 K0.610 59.3

25 469 27.2 K0.090 K1.213 73.061 295 0.91 K0.069 K0.706 70.9

39 455 9.19 K0.074 K1.001 97.6

89 611 1.08 K0.099 K0.857 79.0

18 654 0.45 K0.089 K0.755 103.477 702 0.20 K0.102 K0.655 85.7

68 983 4.25 K0.107 K1.066 107.0

94 776 2.57 K0.095 K0.987 108.1

39 1231 0.26 K0.122 K0.806 98.288 989 6.81 K0.140 K1.198 53.9

49 776 0.27 K0.091 K0.742 117.5

29 835 0.17 K0.096 K0.644 114.9

fatigue tests conducted in this study. All dimensions are in mm.

urnalthe knife-edges of the extensometer, epoxy coating was used

to cushion the attachment. Strain control was used in all tests,

except for some long-life and run-out tests (i.e. where little or

no plastic deformation exits), which were conducted in load-

control mode. For these tests, strain control was used initially

to determine the stabilized load. Then load control was used

for the remainder of the test. For the strain-controlled tests,

the applied frequencies ranged from 0.1 to 2.2 Hz, depending

on the applied strain amplitude. For the load-controlled

tests, the frequency was increased to up to 30 Hz in order to

shorten the overall test duration. All tests were conducted

using a triangular waveform. Test data were automatically

recorded at regular intervals throughout each test, and stable

stress and strain amplitude data at about midlife were used

to generate the fatigue properties.Fig. 1. Specimen geometry and dimensions used for monotonic tension and

A. Fatemi et al. / International Jo10422. Bi-linear SN model and its effects on strain-life

and stressstrain curves

2.1. Bi-linear SN model and its difference with the linear

SN model

To improve the stress-life (SN) as well as the strain-life

(3KN) fatigue models for aluminum alloys, a bi-linearloglog sa versus 2Nf representation of the data for each

material is used. The data for a given material are divided

into two regions, I and II. Region I includes the fatigue data

for the plastic dominated life regime, whereas region II

includes the fatigue data for the elastic dominated life

regime. A loglog linear least square fit of the data is used

for each region. Fatigue properties for the bi-linear model

for the two regions are given as s0f1, b1, s0f2, and b2. Fig. 2

shows schematic representations of sa versus 2Nf for both

linear and bi-linear models, including the material proper-

ties obtained based on each model. In this figure, the life atwhich the two lines for regions I and II of the bi-linear

model intersect is called the separation fatigue life (2Ns),

with the true stress amplitude at that life denoted by Dss/2.Experimental sa versus 2Nf data as well as linear and

bi-linear fits for four typical Al alloys, out of the 14 alloys

considered, are shown in Fig. 3. As can be seen from this

figure, the commonly used linear fit represented by Eq. (1) is

not satisfactory for any of these alloys. The bi-linear model

provides much better fit for the experimental data for these

alloys. Eq. (1) can, therefore, be replaced by the following

equations:

Ds

2Z s0f12Nfb1 for region I; 2Nf %2Ns (4)

Ds

2Z s0f22Nfb2 for region II; 2Nf R2Ns (5)

Bi-linear loglog SN fatigue properties of the 14

aluminum alloys are given in Table 2. The differences

2 Nf (log scale)2 Ns

b1

b2

1

I II

b

'12

'f 'f1 s/2

a (log scale) LinearBi-linear

Fig. 2. Schematic relationship of sa versus 2Nf for linear and bi-linear

models.

of Fatigue 27 (2005) 10401050between fatigue lives based on bi-linear model versus

fatigue lives based on linear model for these alloys are

shown in Fig. 4. In this figure, stress amplitudes for the bi-

linear model are calculated from Eqs. (4) or (5) at fatigue

lives of 103, 104, 105, 106, and 107 reversals, while for the

linear model fatigue lives are calculated from Eq. (1) by

using the corresponding stress amplitudes calculated from

the bi-linear model. It can be seen that the differences

between the fatigue lives based on the two models can be

significant at both short and long lives for most Al alloys.

Plot of the separation life fatigue strength Dss/2 versuscyclic yield strength, S0y, of the Al alloys considered isshown in Fig. 5. The cyclic yield strength is obtained from

the cyclic stressstrain curve using an offset method at 0.002

plastic strain amplitude, analogous to the monotonic yield

strength obtained from a tensile stressstrain curve. As can

be seen from this figure, a good correlation exists,

represented by Dss/2Z0.92 S0y. Therefore, for stress

amplitudes larger than 92% of the cyclic yield strength of

A. Fatemi et al. / International Journal of Fatigue 27 (2005) 10401050 1043a material region I behavior can be expected, while at lower

stress amplitudes region II behavior can be expected.

2.2. The effect of bi-linear loglog SN model

on strain-life curve

Eq. (3) relates true strain amplitude to fatigue life. True

strain amplitude versus reversals to failure plots for four

typical aluminum alloys are shown in Fig. 6, showing

Fig. 3. True stress amplitude versus reversals to failure for typical aluminum all

Table 2

Summary of bi-linear loglog stress-life fatigue properties as defined in Fig. 2

Alloy s0f1 (MPa) s0f2 (MPa) b1 b2 2N

6063 478 603 K0.088 K0.113

A356-T6 390 1114 K0.046 K0.156 1

6260 357 654 K0.059 K0.115 56063 247 388 K0.050 K0.089 10

5754-NG 339 1273 K0.041 K0.163 5

6082 419 679 K0.049 K0.107

AlMg4.5Mn 526 719 K0.060 K0.0965456-H311 559 1233 K0.056 K0.149

7475-T761 839 2329 K0.084 K0.184 2

7075-T6 556 985 K0.048 K0.113

7075-T651 706 1276 K0.042 K0.1257075-T7351 921 1003 K0.128 K0.143

2014-T6 661 1860 K0.059 K0.163 2

2024-T3 651 983 K0.057 K0.112the effect of bi-linearity on strain-life curves. To incorporate

the bi-linearity effect, similar to what Stephens and Koh did

for A356-T6 cast aluminum alloys [6], Eq. (3) is replaced

by:

3a Zs0f1E

2Nfb1 C30f2Nfc for 2Nf %2Ns (6)

3a Zs0f2E

2Nfb2 C30f2Nfc for 2Nf R2Ns (7)

oys considered. (a) A356-T6; (b) 5754-NG; (c) 7075-T6 and (d) 2014-T6.

s Dss/2 (MPa) b2/b s0f1=s

0f s

0f2=s

0f

8118 217 1.05 0.86 1.08

2997 253 1.34 0.59 1.67

0238 189 1.28 0.76 1.40

1682 140 1.29 0.84 1.32

0157 218 2.20 0.75 2.80

3862 280 1.09 0.69 1.11

5741 314 1.08 0.80 1.10

5179 345 1.47 0.80 1.76

5868 359 1.72 0.85 2.37

5806 369 1.19 0.72 1.27

1210 524 1.03 0.57 1.04

312 442 1.01 0.93 1.01

1345 366 1.79 0.85 2.40

1823 424 1.17 0.78 1.18

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08

2Nf based on linear model

2Nf

base

d on

bi-l

inea

r mod

el

6063A356-T6626060635754-NG6082ALMg4.5Mn5456-H3117475-T7617075-T67075-T6517075-T73512014-T62024-T3

A. Fatemi et al. / International Journal1044Stephens and Koh [6] used D3/2Z0.004 as the cut-offbetween the two bi-linear regions. In this study, however,

according to Eqs. (6) and (7), the cut-off is the separation

life Ns. This life is obtained from the intersection of the two

bi-linear fits and can vary for different Al alloys.

As can be seen from Fig. 6, bi-linear stress-life behavior

has little effect on total strain-life curve in the low cycle

regime. The difference between linear and bi-linear models

can become significant in the high cycle regime, however,

due to the fact that this region is mainly governed by the

elastic strain-life line (i.e. SN line).

The difference between the two models is also

Fig. 4. Fatigue life based on bi-linear SN model versus fatigue life based

on linear SN model for 14 Al alloys considered.evaluated in the SmithWatsonTopper (SWT) parameter

(E3asmax

p, where smaxZsa for fully reversed loading)

plot. This parameter is often used to incorporate mean

s / 2= 0.92 S'yR2 = 0.8816

0

200

400

600

0 200 400 600S'y , MPa

s/2

, MPa

Fig. 5. Correlation of separation life fatigue strength Dss/2 with cyclic yield

strength S0y.Zsa

EC

sa

K 0 1=n0

(9)

For the aluminum alloys used in this study, K 0 and n 0 aregiven in Table 3. Note that the cyclic yield strength of each

alloy is obtained from Eq. (8) by substituting 0.002 for the

plastic strain amplitude and using K 0 and n 0 for the alloy, i.e.S0yZK 00:002n0 .

Alternatively, K 0 and n 0 can be calculated from strain-controlled fatigue properties using [1]:

K 0 Zs0f

30fb=c(10)

n0 Zb

c(11)

Eqs. (10) and (11) are derived from compatibilitystress effect, as well as to characterize notched (local)

fatigue behavior (i.e. Neubers parameter). Fig. 7 shows

SWT parameter versus reversals to failure for both linear

and bi-linear models for the same four aluminum alloys

shown in Figs. 3 and 6. For the linear model, at a given

life Nf, SWT parameter is calculated by first finding saand 3a from Eqs. (1) and (3), respectively, using the

material properties listed in Table 1 for each alloy. Then,

the expressionE3asa

pis calculated for Nf. For the bi-

linear model, sa and 3a for a given life Nf are calculated

from Eqs. (4) and (6) if Nf%Ns, and from Eqs. (5) and (7)if NfRNs, using the material properties listed in Tables 1and 2. Note that since mean stress tests were not

conducted, Fig. 7 shows the difference between linear

and bi-linear models when used in conjunction with the

SWT parameter, rather than showing mean stress effects.

2.3. The effect of bi-linear loglog SN model on cyclic

stressstrain curve and properties

The cyclic stressstrain curve reflects the resistance of a

material to cyclic deformation. Similar to the monotonic

deformation in a tension test, a plot of true stress amplitude

versus true plastic strain amplitude in loglog coordinates

for most metals including aluminum alloys results in a linear

curve represented by [1]

sa Z K0 D3p

2

n0(8)

where K 0 and n 0 are the cyclic strength coefficient and cyclicstrain hardening exponent, respectively. Substituting plastic

strain amplitude obtained from Eq. (8) into total true strain

range equation results in the cyclic stressstrain curve

represented by RambergOsgood equation:

3a ZD3

2Z

D3e2

CD3p2

ZDs

2EC

Ds

2K 0

1=n0

of Fatigue 27 (2005) 10401050between Eqs. (1), (2) and (8). Values of K 0 and n 0 obtained

+70

1

True

Stra

in A

mpl

itude

, /

2, %

+70

1

True

Stra

in A

mpl

itude

, /

2, %

(d)

(b)

m all

urnal0.10%

1.00%

1E+2 1E+3 1E+4 1E+5 1E+6 1EReversals to Failure, 2Nf

True

Stra

in A

mpl

itude

,

/2, % A356-T6 Aluminum

Linear model Bi-linear model

0.10%

1.00%

1E+2 1E+3 1E+4 1E+5 1E+6 1EReversals to Failure, 2Nf

True

Stra

in A

mpl

itude

, /

2, %

7075-T6 Aluminum

Linear model Bi-linear model

(c)

(a)

Fig. 6. True strain amplitude versus reversals to failure for typical aluminu

A. Fatemi et al. / International Jofrom direct fitting of the experimental data and calculated

from the relations in Eqs. (10) and (11) can be very similar

or very different, depending on the goodness of linearized

fits represented by Eqs. (1), (2) and (8) [12]. A large

difference for aluminum alloys indicates that the elastic

strain-life behavior is not well represented by loglog

linearized fits. For the aluminum alloys investigated, large

differences between the two sets of K 0 and the two sets of n 0

values can be seen in Figs. 8(a) and 9(a), respectively.

If K 0 and n 0 values are calculated from Eq. (4) based onbi-linear fatigue properties s0f1 and b1 in region I, then bysubstituting these properties into Eqs. (10) and (11) we

obtain:

K 0 Zs0f1

30fb1=c(12)

n0 Zb1c

(13)

Region I of the bi-linear relation (Eq. (4)) is used since

this region represents the region with significant plastic

strain, and therefore, the plastic strain term in Eq. (9) with

K 0 and n 0 as material properties. Values of K 0 and n 0

obtained from Eqs. (12) and (13) as compared to the values

obtained from direct fit of the experimental data are shown

in Figs. 8 and 9. Comparisons of Fig. 8(a) with (b) for K 0 andFig. 9(a) with (b) for n 0, show that correlation of K 0 and n 0

values from direct experimental fit of data with those from.10%

.00%

1E+2 1E+3 1E+4 1E+5 1E+6 1E+7Reversals to Failure, 2Nf

Linear model Bi-linear model

5754-NG Aluminum

.10%

.00%

1E+2 1E+3 1E+4 1E+5 1E+6 1E+7Reversals to Failure, 2Nf

Linear model Bi-linear model

2014-T6 Aluminum

oys considered. (a) A356-T6; (b) 5754-NG; (c) 7075-T6 and (d) 2014-T6.

of Fatigue 27 (2005) 10401050 1045compatibility equations are far better based on region I of

the bi-linear fit (i.e. Eqs. (12) and (13)), than based on the

linear fit (i.e. Eqs. (10) and (11)).

Values of K 0 and n 0 obtained from the compatibilityequations are also listed in Table 3. Note that the differences

between K 0 values as calculated from Eqs. (10) and (12), andbetween n 0 values as calculated from Eqs. (11) and (13) arelarge for many of the alloys listed in Table 3. For example,

for the A356-T6 alloy, K 0 values from Eqs. (10) and (12)listed in Table 3 are 1046 and 465 MPa, respectively.

Values of n 0 for this alloy from Eqs. (11) and (13) are listedas 0.191 and 0.075, respectively. These represent a

difference of 125% between the K 0 values and a differenceof 155% between the n 0 values. Implications of these largedifferences between the compatibility equations values

based on linear versus bi-linear fits on life predictions are

presented in Section 3.

Cyclic true stress amplitude versus true strain amplitude

curves for the four typical aluminum alloys are shown in

Fig. 10. This figure shows that the cyclic true stress

amplitude versus true strain amplitude curves obtained

based on K 0 and n 0 from Eqs. (12) and (13) are the same orvery close to the curves based on direct fitting of

experimental data. On the other hand, the curves based on

K 0 and n 0 from Eqs. (10) and (11) can be much differentfrom the curves based on direct fitting of experimental data.

The difference between linear and bi-linear models in this

figure is more pronounced at high stress amplitudes where

100

1000

1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Reversals to Failure, 2Nf

SWT

para

met

er, M

Pa

LinearRegion IRegion IIFatigue Data

100

1000

1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Reversals to Failure, 2Nf

SWT

para

met

er, M

Pa

LinearRegion IRegion IIFatigue Data

100

1000

1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Reversals to Failure, 2Nf

SWT

para

met

er, M

Pa

LinearRegion IRegion IIFatigue Data

100

1000

1E+2 1E+3 1E+4 1E+5 1E+6 1E+7

Reversals to Failure, 2Nf

SWT

para

met

er, M

Pa

LinearRegion IRegion IIFatigue Data

(d)(c)

(a) (b)

Fig. 7. SWT parameter versus reversals to failure for typical aluminum alloys considered. (a) A356-T6; (b) 5754-NG; (c) 7075-T6 and (d) 2014-T6.

Table 3

Comparisons of K 0 and n0 from direct least square fit and from compatibility equations based on linear and bi-linear fits

Alloy From experimental data fit From compatibility equations based on

linear model

From compatibility equations based on

bi-linear model

n 0 K 0 (MPa) n 0Zb/c K 0Zs0f =30fb=c(MPa)

n 0Zb1/c K 0Zs0f1=30fb1 =c(MPa)

6063 0.067 384 0.129 578 0.106 494

A356-T6 0.063 430 0.191 1046 0.075 465

6260 0.047 301 0.074 367 0.048 304

6063 0.068 245 0.097 298 0.070 249

5754-NG 0.032 294 0.074 386 0.041 310

6082 0.051 397 0.115 605 0.057 417

AlMg4.5Mn 0.125 693 0.118 719 0.079 561

5456-H311 0.084 636 0.155 901 0.086 641

7475-T761 0.059 675 0.100 850 0.078 749

7075-T6 0.045 521 0.096 709 0.048 532

7075-T651 0.074 852 0.151 1506 0.052 756

7075-T7351 0.094 695 0.117 790 0.107 750

2014-T6 0.072 704 0.123 912 0.080 734

2024-T3 0.109 843 0.149 1082 0.088 759

A. Fatemi et al. / International Journal of Fatigue 27 (2005) 104010501046

strain-life and stressstrain curves simultaneously, rather

than the effect on each curve separately.

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000 1200 1400 1600K' from compatibility eq.(10), MPa

K' f

rom

leas

t sq.

fit,

MPa

0

200

400

600

800

1000

1200

1400

1600

0 200 400 600 800 1000 1200 1400 1600K' from compatibility eq. (12), MPa

K' f

rom

leas

t sq.

fit,

MPa

(a) (b)

Fig. 8. K 0 from least square fit versus K 0 from compatibility equation based on (a) linear method (b) bi-linear method.

A. Fatemi et al. / International Journal of Fatigue 27 (2005) 10401050 1047To investigate the bi-linearity effects on life predictions,

a notched component with KtZ2.5 is considered. Thestrain-life (i.e. local strain) approach based on Neubersplastic strains are significant. Therefore, the compatibility

equations based on the bi-linear model are more realistic for

Al alloys, as compared to those based on the linear model.

3. Effect of bi-linearity on life prediction of aluminum

alloys

Fatigue failures in components typically initiate from

notches or at stress concentrations. At such locations, local

plastic deformation usually exists and fatigue life is

controlled by both local stress and strain amplitudes. A

parameter incorporating both stress and strain amplitudes is

Neubers rule. This rule relates notch stress and strain to

nominal stress and strain, and is the most widely used notch

stressstrain analysis model used for life prediction of

notched members [1]. The use of Neubers rule in this work

allows investigation of the bi-linear SN model effects onrule is used. Application of the strain-life approach for

0.01

0.03

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19n' from compatibility eq. (11)

n' f

rom

leas

t sq.

fit

(a) (

Fig. 9. n 0 from least square fit versus n0 from compatibility eqthe component involves two steps. First, it requires

determination of local (notch) stresses and strains based

on Neubers rule. Life prediction can then be made using the

local stresses and strains, based on the strain-life equation.

For Al alloys considered, strain amplitudes were obtained at

five lives (103, 104, 105, 106, and 107 reversals) from either

Eqs. (6) or (7), as appropriate, based on bi-linear model.

Next, local stress amplitudes were obtained from Eq. (9)

based on bi-linear model, where K 0 and n 0 used in theequation are obtained from Eqs. (12) and (13), respectively.

Nominal stress amplitudes, Sa, based on Neubers rule were

then obtained from:

3asa ZKtSa2

E(14)

The ratio of Sa with respect to S0 was calculated for each

Al alloy to ensure nominal elastic behavior. For all the

materials considered, this ratio at the highest load level (i.e.

corresponding to a life of 103 reversals) was less than 0.85.

Using nominal stress amplitudes based on bi-linear model

and Eq. (14), local stress amplitudes based on the linearmodel and Neubers rule were then obtained by combining

0.01

0.03

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19n' from compatibility eq. (13)

n' f

rom

leas

t sq.

fit

b)

uation based on (a) linear method; (b) bi-linear method.

(urnal0

50

100

150

200

250

300

350

400

True

Stre

ss A

mpl

itude

, /

2 (M

Pa)

DataBased on K' and n' from eq. (8)

Based on K' and n' from eqs. (12) and (13)Based on K' and n' from eqs. (10) and (11)

(a)

A. Fatemi et al. / International Jo1048Eqs. (14) and (9) to obtain [1]:

s2a

ECsa

sa

K 0 1=n0

ZKtSa2

E(15)

Note that K 0 and n 0 used in Eq. (15) are obtained fromEqs. (10) and (11), respectively. Once local stress

amplitude, sa, is obtained from Eq. (15), 3a is then

calculated from Eq. (9), and substituted into Eq. (3) to

find the corresponding life based on the linear model.

Fig. 11 shows fatigue life based on the bi-linear model

versus fatigue life based on the linear model. The

differences between fatigue lives based on the two models

range between factors of 2 to 3 at shorter lives, to more

than an order of magnitude at longer lives. At longer lives

the difference between predicted lives based on the two

models increases, mainly due to increased error in

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

True Strain Amplitude, /2 (%)

0

50

100

150

200

250

300

350

400

450

500

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

True Strain Amplitude, /2 (%)

True

Stre

ss A

mpl

itude

, /

2 (M

Pa)

DataBased on K' and n' from eq. (8)Based on K' and n' from eqs. (10) and (11)Based on K' and n' from eqs. (12) and (13)

((c)

Fig. 10. True stress amplitude versus true strain amplitude for typical aluminum a0

50

100

150

200

250

300

True

Stre

ss A

mpl

itude

, /

2 (M

Pa)

DataBased on K' and n' from eq. (8)Based on K' and n' from eqs. (10) and (11)Based on K' and n' from eqs. (12) and (13)

b)

of Fatigue 27 (2005) 10401050extrapolation of the linear model strain-life curve to

long lives. This error results from the use of the linear

model at long lives and is on the non-conservative side.

Note that since many components and structures are

designed for long lives (i.e. lives longer than 106 cycles),

the use of the linear model can result in substantial over-

estimation of the fatigue life, which can lead to premature

fatigue failure. This is due to the fact that, as stated

earlier, the difference between the linear and bi-linear

model life predictions increases at longer lives, with the

linear model over-estimating fatigue lives. This is

particularly important in light of the fact that for

aluminum alloys, the so called fatigue limit or endurance

limit occurs at much longer lives (i.e. typically in the

range of 108109 cycles), as compared to steels (typically

around 106 cycles).

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

True Strain Amplitude, /2 (%)

0

50

100

150

200

250

300

350

400

450

500

0.0% 0.2% 0.4% 0.6% 0.8% 1.0% 1.2%

True Strain Amplitude, /2 (%)

True

Stre

ss A

mpl

itude

, /

2 (M

Pa)

DataBased on K' and n' from eq (8)Based on K' and n' from eqs. (10) and (11)Based on K' and n' from eqs. (12) and (13)

d)

lloys considered. (a) A356-T6; (b) 5754-NG; (c) 7075-T6 and (d) 2014-T6.

amplitudes were obtained at five lives (10 , 10 , 10 , 107

1.E+04on e

xp

5754-NG

urnal of Fatigue 27 (2005) 10401050 10494. Prediction of bi-linear fit constants from linear

fit constants

Since strain-life fatigue properties reported in the

literature are generally based on linear fits of the data (i.e.

s0f and b in Eq. (1)), it is desirable to convert these propertiesto bi-linear properties (i.e. s0f1, b1, s

0f2, and b2 in Eqs. (4) and

(5)). Of course if the raw fatigue data are available, these

properties can be directly obtained from bi-linear fits of the

data (i.e. as in Fig. 3). Such raw data, however, are not

generally reported. Therefore, in this section approxi-

mations of the bi-linear fatigue constants based on the

commonly reported data for Al alloys are discussed.

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10

2Nf based on linear model

2Nf

base

d on

bi-l

inea

r mod

el

6063A356-T6626060635754-NG6082ALMg4.5Mn5456-H3117475-T7617075-T67075-T6517075-T73512014-T62024-T3

Fig. 11. The effect of bi-linear model versus linear model on life prediction

of 14 aluminum alloys considered.

A. Fatemi et al. / International JoAs can be seen from Table 3, the bi-linear slope b1 has a

narrow range between K0.041 and K0.06 for 11 of the 14materials, with an average value of K0.05. Therefore, the bi-linear slope b1 can be regarded as a universal constant for Al

alloys. The ratio of bi-linear slope b2 to linear slope b, b2/b,

also has a relatively narrow range between 1.01 and 1.47 for 11

of the 14 materials as shown in Table 3, with an average value

of about 1.2. Therefore, b2 can be estimated as b2Z1.2b.For all the 14 aluminum alloys considered, the ratio of bi-

linear constant s0f1 to the linear constant s0f , has a relatively

narrow range between 0.57 and 0.93 as shown in Table 3,

with an average value of about 0.75. Therefore, s0f1 can beestimated from s0f1Z0:75s

0f . Similarly, for 11 of the 14

aluminum alloys the ratio s0f2=s0f1 has a range between 1.01

and 1.76 (see Table 3), with an average value of about 1.3.

Therefore, s0f2 can be estimated from s0f2Z1:3s

0f .

The separation fatigue life can then be estimated from:

2NS Zs0f2s0f1

1= b1Kb2 Z

1:3s0f0:75s0f

1= K0:05K1:2b

Z 3p 1= K0:05K1:2b (16)The first part of this relation is obtained from intersection

of the two lines represented by Eqs. (4) and (5) at the

separation fatigue life (2Ns). The last part of the relation is

found by substituting for s0f1Z0:75s0f and s

0f2 Z1:3s

0f ;

b1ZK0:05; and b2Z1:2b, as shown in Eq. (16).Predictive capability of bi-linear fit constants from linear

fit constants as presented above, can be evaluated from

Fig. 12, for all the materials considered. In this figure,

fatigue lives based on experimentally obtained bi-linear

constants are compared with fatigue lives based on the

estimated bi-linear constants. To generate this figure, stress3 4 5 6

1.E+02

1.E+03

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+082Nf based on estimated bi-linear model

2Nf

base

d Data with inLife factor scatter bands

2 80% 3 90% 5 94%

Fig. 12. Fatigue life based on experimental bi-linear model versus fatigue

life based on estimated bi-linear model for 14 Al alloys considered.1.E+05

1.E+06

erim

enta

l bi-l

inea

r m 6082ALMg4.5Mn5456-H3117475-T7617075-T67075-T6517075-T73512014-T62024-T3odeland

(5)

lin

2N

Eq

pro

b2Zbas

the

are

5.

an

fol

1.1.E+07

1.E+086063A356-T66260606310 reversals, as shown in Fig. 12) from either Eq. (4) or

, as appropriate, based on experimentally obtained bi-

ear properties listed in Table 2 (i.e. s0f1;s0f2, b1, b2, and

s). These stress amplitudes were then used again in

s. (4) or (5), as appropriate, using the estimated bi-linear

perties (i.e. s0f1 Z0:75s0f ; s

0f2Z1:3s

0f , b1ZK0.05,

1.2b, and 2Ns from Eq. (16)) to find the life, 2Nf,

ed on estimated bi-linear model. As can be seen, 80% of

data are within a life factor of G2 and 90% of the datawithin a life factor of G3, indicating good agreement.

Conclusions

Based on the experimental data presented and the

alysis performed for 14 Al alloys considered, the

lowing conclusions can be drawn:

The bi-linear loglog stress amplitude-life (SN) model

provides a better representation of fatigue behavior for

aluminum alloys, as compared with the commonly used

linear loglog model. The differences between the SN

fatigue lives based on linear versus bi-linear models can

be significant at both short and long lives.

2. Fatigue strength at the separation fatigue life, where the

transition between the two regions of bi-linear fits

occurs, was found to be a function of the material cyclic

yield strength. For stress amplitudes larger than 92% of

cyclic yield strength of an Al alloy region I behavior can

be expected, while at lower stress amplitudes region II

behavior can be expected.

3. Bi-linear stress-life behavior generally has little effect on

total strain-life curve in the low cycle regime. However,

the difference between linear and bi-linear models can

become significant in the high cycle regime, due to the

fact that this region is mainly governed by the elastic

strain-life line.

4. Values of K 0 and n 0 obtained from direct experimentalfits of cyclic stressplastic strain data agree far better

with those obtained from compatibility equations based

on region I of the bi-linear fit, than those obtained based

5.

6.

based on experimental bi-linear fits of the data for the 14

Al alloys considered.

References

[1] Stephens RI, Fatemi A, Stephens RR, Fuchs HO. Metal fatigue in

engineering. 2nd ed. New York: Wiley; 2000.

[2] Basquin OH. The exponential law of endurance tests. Am Soc Testing

Mater Proc 1910;10:62530.

[3] Endo T, Morrow J. Cyclic stressstrain and fatigue behavior of

representative aircraft metals. J Mater 1969;4(1):15975.

[4] Sanders Jr TH, Mauney DA, Staley JT. In: Jaffee RI, Wilcox BA,

editors. Strain control fatigue as a tool to interpret fatigue initiation of

aluminum alloys. Fundamental aspects of structural alloy design. NY,

USA: Plenum Publishing; 1977.

[5] Wong WA. Monotonic and cyclic fatigue properties of automotive

aluminum alloys. SAE technical paper no. 840120 1984.

[6] Stephens RI, Koh SK. In: Stephens RI, editor. Bi-linear loglog elastic

strain-life model for A356-T6 cast aluminum alloy round-robin low

cycle fatigue data. Fatigue and fracture toughness of A356-T6 cast

aluminum alloy. SAE SP-760 1998.

A. Fatemi et al. / International Journal of Fatigue 27 (2005) 104010501050Bi-linear fit constants can be estimated from commonly

available linear fit constants. Fatigue lives obtained

based on the proposed estimations agreed well with thoseFor notched members, the difference between fatigue

lives based on linear versus bi-linear models range

between factors of 2 to 3 at shorter lives, to more than an

order of magnitude at longer lives. At longer lives the

difference between predicted lives based on the two

models increases, due to increased error in extrapolation

of the linear model strain-life curve to long lives. The

error resulting from the use of the linear model is on the

non-conservative side.on the linear fit. Compatibility equations based on the bi-

linear model are more realistic for Al alloys, as compared

to those based on the linear model.[7] Wigant CC, Stephens RI. Low cycle fatigue of A356-T6 cast

aluminum alloy. SAE technical paper no. 870096 1987.

[8] Stephens RI, Berns HD, Chernenkoff RA, Indig RL, Koh SK,

Lingenfelser DJ, et al. In: Stephens RI, editor. Low cycle fatigue of

A356-T6 cast aluminum alloya round-robin test program. Fatigue

and fracture toughness of A356T6 cast aluminum alloy. SAE SP-

760 1988.

[9] Wong WA, Bucci RJ, Stentz RH, Conway JB. Tensile and strain-

controlled fatigue data for certain aluminum alloys for application in

the transportation industry. SAE technical paper no. 870094 1987.

[10] Boller CHR, Seeger T. Materials data for cyclic loading-part D:

aluminum and titanium alloys. Material science monographs. New

York: Elsevier; 1987.

[11] Annual Book of ASTM Standards. Metals test methods and analytical

procedures. vol. 03.01. West Conshohocken, PA: ASTM; 2003.

[12] Roessle ML, Fatemi A, Khosrovaneh AK. Variation in cyclic

deformation and strain-controlled fatigue properties using different

curve fitting and measurement techniques. SAE technical paper no.

1999-01-0364 1999.

Application of bi-linear log-log S-N model to strain-controlled fatigue data of aluminum alloys and its effect on life predictionsIntroductionBi-linear S-N model and its effects on strain-life and stress-strain curvesBi-linear S-N model and its difference with the linear S-N modelThe effect of bi-linear log-log S-N model on strain-life curveThe effect of bi-linear log-log S-N model on cyclic stress-strain curve and properties

Effect of bi-linearity on life prediction of aluminum alloysPrediction of bi-linear fit constants from linear fit constantsConclusionsReferences