An evaluation of existing methods for the prediction of axial fatigue life from tensile data

Int J Fatigue 15 No 1 (1993) pp 13-19

An evaluation of existing methods for the prediction of axial fatigue life from tensile data J.H. Ong

In the strain-based approach to fatigue, the fatigue properties of materials are characterized by the curves of strain amplitude versus life reversals, obtained from strain-controlled fatigue testing of smooth specimens. In the absence of experimentally determined values, there are three methods which are commonly used for predicting the fatigue properties from uniaxial tension tests. In this paper, these methods are analysed based on 49 steels. The evaluation showed that the four-point correlation method yields the best estimates among the three techniques. The predicted values by the four-point correlation method and the universal slopes method give satisfactory agreement with experimental data. The prediction analysed on the basis of the report by D.F. Socie et al fairs badly in comparison to the other two methods. Key words: axial fatigue life; tensile data; strain amplitude The use of fatigue information in design has increased steadily over the years. One major factor contributing to this increase is a growing recognition of the importance of obviating fatigue failure, not only to prevent catastrophes, but also to forestall breakdowns whose consequences are mainly economic. Another major factor is the acknowledgement that it is too expensive to eliminate potential fatigue failures by gross overdesign. As a result, fatigue design philosophy has changed from one based on endurance limit approaches to one based upon a more precise assessment of fatigue durability. Consequently, modern methods of fatigue analysis are now utilized at the design stage. One such method is the local strain-life approach 1-7 which is now commonly adopted by most industries for fatigue analysis. This approach combines the measured/design service loads imposed upon the structures at the most highly strained locations and the materials' fatigue properties. The materials' fatigue properties are characterized by the strain-life curves, obtained from strain-controlled fatigue testing of smooth specimens. 8 In 1987, accumulation of fatigue test data on steels, aluminium, titanium alloys, cast and welded metals was published by Elsevier in five volumes entitled Materials Data for Cyclic Loading. 9 Although the published fatigue test data are comprehensive, nevertheless it is still probable that the analyst's required data may not be in the literature. Fatigue testing is a time-consuming process. Hence, for the initial design studies, it is highly desirable to estimate the strain-life curve from a knowledge of the more readily obtained material monotonic properties obtained in simple tensile tests. In this study, three commonly used methods *°'*~ utilized for predicting fatigue strain-life relation from monotonic properties of materials are analysed. The study is based on 49 steels which cover quite a range in variables that might affect fatigue behaviour such as cyclic hardening/softening characteristics, reduction in areas covering the range from 10 to 80% and tensile strengths from 300 to over 2500 MPa.

Strain-life properties Fatigue resistance of metals can be characterized by a strainlife curve as shown in Fig. 1. The total strain amplitude, Ae/2, can be considered as the summation of elastic ( ~ d 2 ) and plastic (A%/2) strain amplitudes as expressed by a ~ _ aEo + a ~ r

2

2

(1)

2

The elastic strain-life relation is simply the stress-life relationship divided by the modulus of elasticity: ~l~r _ Ae~ _ cr[ (2Nr)~ 2E 2 E

(2)

where &r/2 is the stress amplitude; 2Nf is reversals to failure; crf is the fatigue strength coefficient, taken as the intercept of the log (&r/2) versus log (2Nf) plot at 2Nf = 1 (Fig. 1); and b is the fatigue strength exponent, taken as the slope of the log (&r/2) versus log (2N¢) plot (Fig. 1).

I

Elastic

Plastic

f ~1"~

_

o~

b

c

~

~

laTot =1elasticand plastic

lo

3 t-

~=~

~astic

~3

\~:3 Plastic 107

100 Reversals to failure, 2Nf Fig. 1 Typical strain-life curve

0142-1123/93/010013-07 © 1993 Butterworth-Heinemann Ltd Int J Fatigue January 1993

13

The plastic strain-life relationship can be written in terms of

B=½× = e/(2N0 ~

2

(3)

where ~f is the fatigue ductility coefficient, taken as the intercept of the log (A%/2) versus log (2Nf) plot at 2Nf = 1 (Fig. 1) and c is the fatigue ductility exponent, taken as the slope of the log (A%/2) versus log (2Nf) plot (Fig. 1). The resulting strain-life curve (as shown in Fig. 1) can thus be expressed as AE _ (r[ (2Nf)b + e/(2Nf)¢ 2 E

(4)

The four constants o~[,b,a[, and c define the basic fatigue properties of materials. These are listed for the 49 types of steel which are used for the present study in Table 1. The fatigue properties as listed in Table 1 are taken from the ASM Metals Handbook. 12

1 1/(1- RA)}] 3/4 ] l + logic[In{ 10 c1°g56

(10)

E is the elastic modulus.

Universal slopes method An alternative approach was also proposed by" Manson l° which assumes that the slopes of the elastic and plastic lines are the same for all materials. The slopes of the elastic and plastic lines are assumed to be equal to -0.12 and -0.6 respectively. The original equation as proposed by Manson is written in terms of total strain range versus life cycles. Manson 1° shows that the intercept of the elastic line at life cycle Nf = 1 is equal to [In(1 - R A ) - I ] ° 6 and the intercept for the plastic line at N f - - 1 is equal to 3.50yu/E ). The equation for the total strain range Ae then becomes AE= 3 . 5 ~ - ( N f ) -°12 + In i~-_R ~

(NO -°6

(111

Four-point correlation method This method was introduced by Manson 1° and is referred to as the four-point correlation method because the elastic and plastic lines as shown in Fig. 2 are obtained by locating two points on each of them. A point is located at the elastic line at 41 cycle with an ordinate (2.5~q)/E, where ~f is the true fracture stress of the material obtained by dividing the load at the time of failure in the tensile test by the actual area measured after failure has occurred. Another point on this line is obtained at 105 cycles. At this point, the ordinate is (0.9~r,)/E, where Or, is the conventional ultimate tensile strength of the material. On the plastic line, a point at 10 cycles is determined that has an ordinate of ~D3/4, where D = ln(1-RA) -1 is the logarithmic ductility of the material and RA is the reduction in area. The second point of the plastic line is obtained at 10* cycles as indicated in Fig. 2. The point shown by the @ symbol at 104 cycles is first located on the elastic line and the ordinate observed. This ordinate is then substituted into the simple equation shown in the figure to obtain a corresponding ordinate value at 104 for the plastic strain. The formula is derived from the observation that the plastic and elastic strains at 104 cycles are approximately related to each other and the total strain at that point is approximately 1% for all materials.13 Rearranging the four-point correlation method into the now widely accepted form of the strain-life equation as presented in Fig. 1 gives Ae 2 - A(2N0b + B(2N0~

(5)

Rewritten in the widely accepted form of total strain amplitude versus life reversals, the above equation becomes A~ o', o 2 - 1 . 9 0 1 8 ~ ( 2 N f ) - 12 1

Method proposed by Socie et a111 In the report by Socie et al, 11 the fatigue properties of materials are calculated from its monotonic properties based on the following assumptions. 1)

2)

3)

4)

where b = log [2.5[1 + ln{1/(1 R A ) }-] ] 0 . 9 (6)

log[l/(4 × 10s)] A = ½ × l0 b log2 + log[2"5°'u[1

+ 1n{1/(1 RA)}]]

E

- -

=

10blog(4×104)+log[ 25°'u[1+ln(1/(l-RA)}~]

c=½

14

~

Fatigue strength coefficient ¢[ is equal to the true fracture stress ~f corrected for necking. However, the true fracture stress is not always given in the literature and therefore an additional approximation is required. It is suggested that for steels to about 500 Brinell hardness, ~r[ = co + 345 MPa. To obtain the elastic slope b, a point is first located on the elastic line at 106 reversals with an ordinate 0.5Gru. The elastic slope is calculated by joining a straight line between this point and a second point at one reversal with an ordinate equal to the true fracture stress or the ultimate tensile stress plus 345 MPa. Fatigue ductility coefficient ~[ is approximated to be equal to the logarithmic ductility of the material: ie E[ = In(1 - RA) -1. Fatigue ductility exponent c is not as well defined as the other parameters and is assumed to be equal to - 0 . 6 for fairly ductile material and is equal to -0.5 for a strong metal.

From the above assumptions, the strain-life equation becomes

(7)

A ~ is the strain located on the elastic line at 104 cycles (see Fig. 2), given by A~e

0.6

(8)

1og[0.0132, - Ae~ ] _ ~ f l [ / 1 \]3/4~ L 1[91 - ] ~l°g/4[ln~l~Z~)] / (9)

(13) or

A~ - ~r + 345 (2Nr)-~log 1 [2(~u+ 345)/au] 2 E (14)

Int J Fatigue January 1993

Table 1. Monotonic and fatigue properties of selected steels

SAE spec

Brinell hardness

Ultimate strength (MPa)

True fracture stress (MPa)

Reduction in area (%)

True fracture strain

A538A* A538B* A538C* AM-350t H-11 RQC-100t RQC-100t 10B62 1005-1009 1005-1009 1005-1009 1005-1009 1015 1020 1040 1045 1045 1045 1045 1045 1045 1144 1144 1541F 1541F 4130 4130 4140 4142 4142 4142 4142 4142 4142 4142 4142 4340 4340 4340 5160 52100 9262 9262 9262 950C 950C 950X 950X 980X

405 460 480 496 660 290 290 430 90 125 125 90 80 108 225 225 410 390 450 500 595 265 305 290 260 258 365 310 310 380 400 450 475 450 475 560 243 409 350 430 518 260 280 410 159 150 150 156 225

1515 1860 2000 1905 2585 940 930 1640 360 470 415 345 415 440 620 725 1450 1345 1585 1825 2240 930 1035 950 890 895 1425 1075 1060 1415 1550 1760 2035 1930 1930 2240 825 1470 1240 1670 2015 925 1000 1565 565 565 440 530 695

1896 2137 2241 2179 3172 1069 1331 1779 717 745 841 848 724 710 1048 1227 1862 1862 2103 2275 2723 1158 1517 1276 1276 1420 1820 1524 1117 1827 1896 1999 2068 2103 2172 2654 1089 1558 1655 1931 2193 1041 1220 1855 931 1000 752 1000 1220

67 56 55 20 33 43 67 38 73 66 64 80 68 62 60 65 51 59 55 51 41 33 25 49 60 67 55 60 29 48 47 42 20 37 35 27 43 38 57 42 11 14 33 32 64 69 65 72 68

1.10 0.82 0.81 0.23 0.40 0.56 1.02 0.89 1.30 1.09 1.02 1.60 1.14 0.96 0.93 1.04 0.72 0.89 0.81 0.71 0.52 0.51 0.29 0.68 0.93 1.12 0.79 0.69 0.35 0.66 0.63 0.54 0.22 0.46 0.43 0.31 0.57 0.48 0.84 0.87 0.12 0.16 0.41 0.38 1.03 1.19 1.06 1.24 1.15

Fatigue Modulus of strength Elasticity coefficient (GPa) (MPa) 185 185 180 180 205 205 205 195 205 205 200 200 205 205 200 200 200 205 205 205 205 195 200 205 205 220 200 200 200 205 200 200 200 200 205 205 195 200 195 195 205 205 195 200 205 205 205 205 195

1655 2135 2240 2690 3170 1240 1240 1780 580 515 540 640 825 895 1540 1225 1860 1585 1795 2275 2725 1000 1585 1275 1275 1275 1695 1825 1450 1825 1895 2000 2070 2105 2170 2655 1200 2000 1655 1930 2585 1040 1220 1855 1170 970 625 1005 1055

Fatigue Fatigue Fatigue strength ductility ductility exponent coefficient exponent -0.065 -0.071 -0.070 -0.102 -0.077 -0,070 -0.070 -0.067 -0.090 -0.059 -0.073 -0.109 -0.110 -0.120 -0,140 -0,095 -0.073 -0.074 -0.070 -0.080 -0.081 -0.080 -0.090 -0.076 -0.071 -0.083 -0.081 -0.080 -0.100 -0.080 -0.090 -0.080 -0.082 -0.090 -0,081 -0.089 -0.095 -0,091 -0.076 -0.071 -0.090 -0,071 -0.073 -0.057 -0.120 -0.110 -0.075 -0.100 -0.080

0.30 0.80 0.60 0.10 0.08 0.66 0.66 0.32 0.15 0.30 0.11 0.10 0.95 0.41 0.61 1.00 0.60 0.45 0.35 0.25 0.07 0.32 0.27 0.68 0,93 0,92 0,89 1.20 0.22 0.45 0.50 0.40 0.20 0.60 0.09 0.07 0.45 0.48 0.73 0.40 0.18 0.16 0.41 0.38 0.95 0.85 0.35 0.85 0.21

-0.62 -0.71 -0.75 -0.42 -0.74 -0.69 -0.69 -0.56 -0.43 -0.51 -0.41 -0.39 -0.64 -0.51 -0.57 -0.66 -0.70 -0.68 -0.69 -0.68 -0.60 -0.58 -0.53 -0.65 -0.65 -0.63 -0.69 -0.59 -0.51 -0.75 -0.75 -0.73 -0.77 -0.76 -0.61 -0.76 -0.54 -0.60 -0.62 -0.57 -0.56 -0.47 -0.60 -0.65 -0.61 -0.59 -0.54 -0.61 -0.53

*ASTM designation. tTradename: Bethlehem Steel Corp

Comparison of prediction methods Forty-nine materials as listed in Table 1 were used for the analysis. Figures 3-8 present the overall comparisons between the experiments and the predictions when analysed on the basis of each of the three methods described in the preceding sections. The experimental values are calculated from the materials' fatigue properties as listed in Table 1. Figures 3-5 show the experimental life reversals for each of the calculated points plotted against the predicted life reversals from a knowledge of the true fracture stress, ultimate tensile strength and logarithmic ductility. The corresponding data on the basis of experimental and predicted strain amplitudes rather than life reversals are shown in Figs 6-8. The relationship between

Int J Fatigue January

1993

the data points and the 45° line is indicated as a table and is shown in all the figures. Scanning through Figs 3-8, it is observed that the fourpoint correlation method and the universal slopes method are approximately the same. The prediction analysed on the basis of the report by Socie et a l n fairs badly in comparison to the other two methods. When comparing life, the four-point correlation method and the universal slopes method have 86 and 80% of the data points respectively, falling within a life factor of 5 from the experimental values. However, within a life factor of 5, only 59% of the data points predicted on the basis of the report by Socie et al falls within this band and only with an allowable error of a life factor of 20 do 84% of

15

1012 lr~314

1°1° I o >

c

2.5

A~= Aee+Z~c

I08L

Life factor I .5 2.0

% of data 17 32

5.0 10.0 20.0

59 72 84

o~

°° o

o

@o

106 104 o o :~o=OOO , - Oo n

I04

105

o

102 t

10

_

°Oo

100

I

101

I

102

I

103

Cycles to failure, Nf

i

104

105

I

I

106

107

Fig. 2 Method of predicting properties based on the four-point correlation method

Fig. 5 Comparison of predicted and experimental life reversals by method based on Socie et aP ~

0.3

o

08

o

•~ 45° I i_

108

Life, reversals

o

0.1

o

o

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104

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1

o Life

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o oo ~o~y --

102

,

~

1.5

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° ° =~Ir-

o. 10 0

factor

~ of data 38

2.0

s.

5.0 10.0 20.0

86 94 97

I

I

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t

10 2

10 4

10 6

10 8

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Strain factor 1.I 12

_=.mr° ~l~o'e

0.003 .u_

% of data 38 60

o

.j,~ ° ~ ~o

1.5

0.001

84

X

o_

0.0003

0.001 0.003

0.01

0.03

0.1

0.3

Total strain amplitude

Life, reversals Fig, 3 Comparison of predicted and experimental life reversals by the four-point correlation method

Fig. 6 Comparison of predicted and experimental strain amplitudes by the four-point correlation method

0.3 108

o "D

o

o

o°

o

0.1 o

~-

106 r.

~-

L" Life

° ° ~ og9 o o a@0

o

/ ~

._u 102

factor 1.5

3.0

oo~o

J 100

5.0

10.0 20.0 [

I

102

104

I

106

~ of data 36

0.03

v~r/ ,.~lllv

Fig. 4 Comparison of predicted and experimental life reversals by the universal slopes method

Strain factor

~ of data

1.2 1.5 2.0 5.0

57 81 98 100

~=o

.?.

-o o

0.001

/~8#

/

Q.

I

108

--

0.003

so

80 94 98

Life, reversals

16

o

0.01

104

oo

cL

o oo~ oo

0.0003

J

J

I

I

l

I

0.001

0.003

0.01

0.03

0.1

0.3

Total strain amplitude

Fig. 7 Comparison of predicted and experimental strain amplitudes by the universal slopes method

Int J Fatigue January 1993

3500 0.3 = ~J

"o

0.1

~ oo

0.03 r-

0.01 0

0.003

°

o

%°o

E ou

E

1.1

o

3000 o

2500

.~

74

500

97 100 500

0.0003

25=00

i•

1000

16

2.0 5.0

20100

:°/J

lSOO _36

1.5

o

°°oo~°

2000

~ o!_data

1.2

~ °~

o

line

~ °

0.001

~

o o o~ o~O

Strain factor

Ai~

°

o o° °

~

= 07-

u :~

o

u

I

I

l

I

I

I

0.001

0.003

0.01

0.03

0.1

0.3

i 1000

L 1500

= 3000

3500

Fatigue strength coefficient

Total strain amplitude Fig. 8 Comparison of predicted and experimental strain amplitudes by method based on Socie et aP ~

the data points lie within this band. Comparing the fourpoint correlation method with the universal slopes method, Figs 3 and 4 show that the predictions for all the life factor bands of 1.5, 2.0, 5.0, 10.0 and 20.0 are approximately the same. It is expected that a few data points will be associated with poorly behaved materials and hence the results as presented in Figs 3 and 4 are acceptable. Since some scatter in life is expected in fatigue ~ata, it can be seen that the correlations for the four-point c6~relation method and the universal method are satisfactory. Figures 6-8 compare the data on the basis of strain amplitudes. The correlations as might be expected follow the same trend as in life predictions. The correlations are better, with 84 and 81% of the data falling within a factor of 1.5 for the fourpoint correlation method and the universal slopes method respectively.

Correlation between fatigue parameters and predicted values The previous section compares the overall results on the basis of life reversals and strain amplitudes and shows that the four-point correlation method gives effectively the best solution of the three methods for estimating fatigue behaviour in terms of monotonic properties from uniaxial tension tests. Since four fatigue parameters (fatigue strength coefficient, fatigue strength exponent, fatigue ductility coefficient and fatigue ductility exponent) are used to describe a strain-life curve, it will be useful to note the correlations between the predicted and experimental values for the four-point correlation method. Figures 9-12 show the overall comparisons between predicted and experimental values for the fatigue strength coefficient, the fatigue strength exponent, the fatigue ductility coefficient and the fatigue ductility exponent respectively. Figure 9 shows that the fatigue strength coefficient is highly correlated. Figures 10-12 show that the correlations for the rest of the three fatigue parameters are rather poor. As a preliminary study, the average value of fatigue strength exponent b equal to -0.084 is used to modify the equations as presented under the section for the four-point correlation method. The results of the simulation in terms of life reversals and strain amplitudes are shown in Figures 13 and 14 respectively. Unfortunately, the results show that the modification gave poorer results as compared to the original

Int J Fatigue January 1993

Fig. 9 Comparison between predicted and experimental fatigue strength coefficient: four-point correlation method

0.04

E o o e~

0.06

£O~

0.08

~o

o

c-

o

o

45° line

0.10

o

o

o

o

o

o

o

D~

0.I~

u

0.14

0.16

I

I

I

I

I

0.14

0.12

0.10

0.08

0.06

0.04

Fatigue strength component Fig. 10 Comparison between predicted and experimental fatigue strength exponent: four-point correlation method

._o

1.5

u

o

o

o

1.0

o line

~,.~

o

o

•

_~3~°05

o ~

~;~

•-~

O.

0.0

:o

I

l

I

0.5

I .0

I .5

Fatigue ductility coefficient Fig. 11 Comparison between predicted and experimental ductility coefficent: four-point correlation method

17

~.

0.3

4000

0.4

3500

X

0.5

o

o

o

3000

o co Q-

ooo

0.6

o o

o

oOo o

o

o o

o

o

°~ °

o

o

o °o

2000

+

'~

250O1

o

0

:3 0

O

1500 1000

0.9

I

I

I

I

I

0.8

0.7

0.6

0.5

0.4

0.3

I

500 Fatigue ductility exponent

%

108 oo o °o

~o

1000

I

I

I

I

I

1500 2 0 0 0 2500 3 0 0 0 3500 4000 True fracture stress (MPa)

Fig. 12 Comparison between predicted and experimental fatigue ductility exponent: four-point correlation method

Fig. 15 True fracture stress against function of logarithmic ductility and ultimate tensile strength

equations for the four-point correlation method. Nevertheless, it is envisaged that it is possible to fine-tune the prediction models so as to obtain a better estimate of fatigue behaviour from monotonic tensile properties.

oo

% o

106

Concluding remarks o

104

{~

-O

Life factor

% of data

1.5

28

10.0

83

20.0

89

oo .L-~/~ o~i.i~o-

;~t,~o ~-o

.u lO 2

2.0

J 100

39

I

I

I

k

102

104

106

108

Life, reversals Fig. 13 Comparison of predicted and experimental life reversals by modified four-point correlation method (constant fatigue strength exponent = - 0 . 0 8 4 )

0.3

m

o o o°

0.8

e~

"~ ~.

45° line

°o

0.7 0

~

o

D

o

~

0.1 o

o~,° oo o oo

0.03

°o

°

~

o o

'

The analysis shows that the four-point correlation method and the universal slopes method give satisfactory agreement with the calculated experimental data. In both cases, the predicted values are approximately the same. However, when the four basic fatigue parameters (~i, b, El', and c) are analysed, the overall results show that the correlations for ~rf' is good but the other three fatigue parameters are poor. Hence it is envisaged that it is possible to get a better model for predicting the fatigue behaviour from monotonic tensile properties. In order to obtain the fatigue properties for the fourpoint correlation method, the true fracture stress must be known. However, this property is not always given in the literature and therefore an additional approximation is required; as suggested in Ref 10, the true fracture stress could be obtained by multiplying the ultimate tensile strength by the factor (1 + logarithmic ductility). This relation is shown in Fig. 15 and is seen to have a high correlation between fracture stress and the factor (1 + logarithmic ductility). Based on this relationship, the predicted data for the fourpoint correlation method were recomputed. The life factors and strain factors calculated were almost as good as the original set of data predicted based on the true fracture stress of the materials.

line 0.01

e ~

References rain factor

o

0.003

.a~

I.1 1.2

o~

•,

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--~ 0.001 ~ o_

"

~

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0.001 0.003

1.

Ong, J.H. 'Fatigue-life predictions on microcomputers' Computer-Aided Design 19 2 (1987) pp 95-101

2.

Dowling, N.E. 'Fatigue life prediction for complex load versus time histories' Trans ASME, J Eng Mater Technol 105 (July 1983) pp 206-214

3.

Martin, J.F., Topper, T.H. and Sinclair, G.M. 'Computer based simulation of cyclic stress-strain behaviour with applications to fatigue' Mat Res Stand 11 2 (1971) pp 23-29

4.

Landgraf, R.W. and La Pointe, N.R. 'Cyclic stress-strain concepts applied to component fatigue life prediction' SAE paper No. 740280 (Proc. Automotive Engineering Congr., Detroit, Mich., 1974)

47

1.5

/7 ~/

0.0003

% o_fdata

93 94

i

I

I

I

0.01

0.03

0.I

0.3

Measured totalstrainamplitude Fig. 14 Comparison of predicted and experimental strain amplitudes by modified four-point correlation method (constant fatigue strength exponent = -0.084)

18

Int J Fatigue January 1993

5.

Schutz, D. and Gerharz, J.J. 'Critical remarks on the validity of fatigue life evaluation methods based on local stress-strain behaviour' ASTM STP 637 (American Society for Testing and Materials, 1977) pp 209-233

6.

Socie, D.F. 'Fatigue life prediction using local stress-strain

7.

Wetzel, R.M. (ed) Fatigue Under Complex Loading: Analyses and Experiments Vol AE-6 (Society of Automotive Engineers, 1977)

8.

'Standard recommended practice for constant-amplitude low-cycle fatigue testing (E606-80)' Annual Book of ASTM Standards Vol 03.01 (American Society for Testing and Materials, 1989) pp 601-613

9.

Boiler, C. and Seeger, T. Materials Data for Cyclic Loading Parts A to E (Elsevier Science Publishers B.V., 1987)

10.

Manson, S.S. 'Fatigue: a complex subject--some simple approximations', Exp Mech SESA (1965) pp 193-226

11.

Socie, D.F., Mitchell, M.R. and Caulfleld, E.M. 'Fundamentals of modern fatigue analysis' Fracture Control Program Report No. 26 (University of Illinois, USA, 1977)

12.

'Properties and selection: irons and steels' ASM Metals Handbook 9th edn (American Society for Metals, Metals Park, OH, 1978) 1 p 680

13.

Menson, S.S. and Hirschberg, M.H. 'Fatigue behaviour in strain cycling in the low and intermediate cycle range' in J.J. Burke, N.L. Reed and V. Weiss (eds) Fatigue--An /nterdisp/inary Approach (Syracuse University Press, 1964), pp 133-173

concepts' Exp Mech SESA 17 2 (1977) pp 50-56

Int J Fatigue J a n u a r y 1993

Author

The author is with Nanyang Technological University, Nanyang Avenue, Singapore. Received 5 May 1992; accepted in revised form 7 July 1992.

19