- Email: [email protected]
The use of approximate strain-life fatigue crack initiation predictions
The use of approximate strain-life fatigue crack initiation predictions F.P. Brennan
NDE Centre, Department of Mechanical Engineering, University College London, Torrington Place, London WCIE 7JE, UK (Received 25 November 1993) This paper describes the most commonly used approximate strain-life equations for prediction of fatigue crack initiation. Despite the development of advanced multiphase material memory models, the simple strain-life equations are useful 'first-shot' approximations of a component's resistance to fatigue crack initiation. There are, however, certain precautions that should be observed when using these equations. Selection of appropriate cyclic material properties is of utmost importance. These properties must be internally consistent; however, many values given in published literature are not. A sensitivity analysis is also presented for predicted fatigue life with respect to percentage error in cyclic material properties. (Keywords: strain Hfe; cyclic material behaviour; consistency; sensitivity; engineering design)
Engineering experience has adopted a general empirical approach to crack generation that does not consider the evolution of individual microcracks, but defines initiation in terms of bulk material properties and the local stress system. The cyclic nature of material can vary considerably from its static stress-strain behaviour. The reasons for the differing behaviour are similar to the change in material characteristics due to cold working. Materials can harden or soften in the same cyclic load sequence. The cyclic behaviour stabilizes after a number of cycles (usually quite quickly), this achievement of a convergent cyclic behaviour is known as saturation hardening. If the cyclic stress-strain behaviour of a reversed load situation is plotted directly, the result is a hysteresis loop. This hysteresis loop stabilizes after a number of cycles on realization of saturation hardening for a given strain range. Definition of the cyclic stress-strain curve can be taken as the locus of tips of the stable hysteresis loops from several tests at different completely reversed constant strain amplitudes. Knowing the cyclic behaviour of a material, it is possible to estimate degradation of the material under repeated cyclic loading. Recently there has been renewed interest in approximate empirical strain-life fatigue crack prediction methods. These first appeared 30 years ago and are still widely used as 'first shot' indications of fatigue resistance or for material optimization studies. This paper aims to remind designers of the origin of the expressions, and tries to establish the degree of accuracy that should be expected from these expressions in predicting fatigue crack initiation.
F A T I G U E LIFE FROM M A T E R I A L S D A T A The concept that cyclic life was related to plastic strain was proposed by Coffin and Manson; however, it is usually the total strain that is known rather than just the plastic component. The total strain range is a simple summation of elastic and plastic components, but unlike its constituent parts does not have a relationship to the number of cycles to failure that is a straight line on logarithmic axes. The total strain range-fatigue life curve is asymptotic to the plastic line in the low-cycle regime, and to the elastic line in the high-cycle regime (Figure 1). Manson 1 detailed two methods whereby a material's relationship of total strain range to fatigue life can be evaluated from simple tension tests. The first of these is the fourpoint correlation method. As the relationships of elastic and plastic strain components to life are
Total StrmnLine 0og) Stma
RJmge
(t)
(log) Cycles to Failure
Figure 1 Strain-life diagram 0142-1123/94/050351-06 © 1994 Butterworth-Heinemann Ltd
Fatigue, 1994, Vol 16, J u l y
351
Approximate strain-life fatigue crack initiation predictions: F.P. Brennan approximately straight lines, only two points are required to define each case. From observation of material behaviour the following points were suggested. Elastic line: (1) N f = l / 4 , (2) Nf = 105 Plastic line: (1) Nf = 10 (2) Uf = 104
e¢=2.5crf E I~ e
= 0.9 OUTS E
Ae _ ~ (2Nf)b + e; (2Nf)c 2 E
i~p ~---0.25 D 3/4 0.0132 - ee ep 1.91
= fatigue strength b = fatigue strength e'f = fatigue ductility c = fatigue ductility
(ductility) (true fracture stress)
D)
and RA is the percentage reduction in area due to waisting of the tensile specimens. By using this approximation, only the elastic modulus and two tensile properties (the ultimate tensile strength and reduction in area) are needed to predict axial fatigue life for a specified local strain range. The other approach is to assume that the slopes of the elastic and plastic lines are the same for all materials. Again, Manson 1 and Hirschberg studied the behaviour of 29 materials, including body-centred cubic, face-centred cubic, hexagonal close packed, precipitation hardened, hot and cold worked, notch ductile, notch sensitive, strain softening, strain hardening and displaying a large variation in tensile strength, elastic modulus and ductility. They presented the following equation as a reasonable representation of the total strain range relationship with cycles to failure: AI3 -- 3"5OrUT~S N f ° ' 1 2
+ O°6Nf
°'6
(1)
E The prominence of this equation in design standards and practices is due to its simplicity and application to many different types of metal. In a more recent study 2, Muralidharan and Manson, using monotonic and fatigue data for 47 materials including steels, aluminium and titanium alloys, derived a modified universal slopes equation for the estimation of fatigue life characteristics of metals. This is given as:
The first term represents the elastic strain component; the second, the plastic strain component. Knowing these material constants, the strain-life relationship can be evaluated, and then used to predict the onset of fatigue cracking. Unfortunately, these material properties are often not readily available. The following sections highlight the significance of these, their characteristics and methods for their determination. FATIGUE DUCTILITY PROPERTIES Morrow 4 proposed the form of the material properties and Equation (3), which have since been adopted as standard. Unlike the previous approximate relations where Manson et al. predicted the material cyclic behaviour from simple tensile test results, these empirical fatigue properties are established from a series of completely reversed axial fatigue tests carried to fracture on smooth specimens. From a series of fatigue tests at different strain ranges, a plot of the log of stable plastic strain amplitude against the log of fatigue life produces a series of points that closely fits a straight line. The slope of the line will vary depending on the material. The plastic strain intersection at one half cycle (one reversal) determines the level of fatigue ductility, i.e. the fatigue ductility coefficient e~. It can be approximated by 5 / ~ \ira,
)
(4)
N ( °-56
+ 1.17(~-)0"832 N f 0.09
(2)
The study showed that the influence of ductility on the elastic line is negligible. However, it is strongly influenced by the ratio of tensile strength to modulus of elasticity. Ductility is less important than indicated by the earlier universal slopes relation for plastic strain. The ductility exponent was 0.6 but is modified to 0.155. The exponent -0.12 has been revised to -0.09 for the elastic strain component, implying greater fatigue resistance as extrapolations are made to lives much greater than 106 cycles. However, before adoption of either Equation (1) or Equation (2) the particular material should be researched with regard 352
coefficient; exponent; coefficient; and exponent.
- O--ys
/(7" TS \--0.53
Ae = O.O266DO.1551~- )
(3)
where:
where
O'f ~" O'UTS(1 +
to likely fatigue behaviour. For example, the modified universal slopes equation is less accurate for AISI 4130-4340 steels below 104 cycles but more accurate than the old equation in the high-cycle regime. An important use of these relationships is to indicate the sensitivity of tensile strength and ductility on fatigue life. Ideally, relationships should be obtained for particular materials. The most common form is given as3
Fatigue, 1994, Vol 16, July
where ~r~.s is the cyclic 0.2% offset yield strength, i.e. the stress that causes 0.2% inelastic strain on the cyclic stress-strain curve. The fatigue ductility coefficient is the true strain required to cause failure in one reversal, i.e. the intersection of the line /Aep\ log~ ~ - ) - log 2Nf
(5)
at 2Nf = 1. It is related to the monotonic fracture ductility ef, which is defined as the 'true' plastic strain required to cause fracture:
I~f= ln(lO0 ZO/oRA 100 1]
(6)
Approximate strain-/ife fatigue crack initiation predictions: F.P. Brennan The true plastic fracture strain is the same value as Manson's ductility D. The slope of the plastic strain-life line may also be considered a fatigue property and is called the fatigue ductility exponent c. This is nearly a constant for metals and normally has a value between -0.5 and -0.7, with - 0 . 6 as a representative value. It is reasonably independent of chemical composition, hardness and temperature. The plastic strain-life relationship defined in terms of the ductility properties, Aep 2 - e;(2Nf)c
(7)
documented were chosen. The mechanical properties of these are shown in Table 1. The values given are taken from ref. 7, which includes many other steels; however, the six chosen were judged to be representative of a broad range of steels, and the cyclic material properties were self-consistent. Many published material properties do not exhibit self-consistency and must therefore be regarded with scepticism. The consistency can be checked by the relations b n' = -
(12)
c
is similar to the Manson-Coffin relationship except that plastic strain amplitude is used and reversals instead of cycles. FATIGUE STRENGTH PROPERTIES The first term in Equation (3) represents the elastic strain component. The fatigue strength coefficient can be related to the plastic strain component by4 acp
1'"'
2
\K']
,
= e, \ 4 1
= e; (2Ne)c
(8)
Letting n' c = b o-, = 4 (2N,) b
(9)
so
2Nf
=[trall/b \4]
(10)
The same expressions can be found by plotting the log of stress amplitude against the log of fatigue life. The fatigue strength coefficient 4 becomes the 'true' stress required to cause failure in one reversal, i.e. the intersection of the line log ~
- log 2Nf
(11)
at 2Nf = 1. It is proportional to the true fracture strength ( 4 ~ ~f) and ranges between about 690 MPa and 3400 MPa for heat-treated metals 6. The fatigue strength exponent is the slope of the line given by Equation (11). It normally is a maximum of -0.12 for softened metals and decreases to a minimum value of about -0.05 with increasing hardness. The strength properties are also referred to as 'Basquin' constants after his 'exponential law' of fatigue (1910), which is similar to Equation (9). It can be appreciated that only four of the six cyclic material constants discussed are unique; the remaining two can be derived from the four others. DISCUSSION The previous sections have set out the most established strain-life engineering fatigue crack initiation models, and have detailed various approximations and assumptions frequently employed. It is a useful exercise to study the implications of adopting approximations, especially where it is necessary to know if a fatigue life prediction is conservative, and what the approximate degree of conservatism is, or otherwise. In order to do this, six steels whose cyclic material properties are
4 K' -
(13)
A certain percentage error (up to 20%) is often tolerable, depending on the number of decimal places to which each value is reported. For example, in Table 1, n' is only given to two decimal places. Values of 0.104 and 0.095 would both be given as 0.10, but the lowest value is 8.7% less than the largest. However, in the literature it is quite common to find errors of 50-100% when the above checks are applied. Most discrepancies in material values arise from inaccurate regression or data analysis8; or too many data points are frequently considered from one test specimen, leading to adoption of data points before saturation hardening is achieved. Equations (12) and (13) are useful ways of assessing confidence in cyclic material data. Fatigue life predictions using Equation (3) were compared with the four-point correlation method and the two universal slopes approximations (Equations (1) and (2)) for the six steels. The four-point correlation method was used by employing a closedform solution devised by Ong 9. Figure 2 shows the prediction as a line having a 45 ° slope. For each material the number of cycles required to give an equal total strain range given by Equation (3) was calculated by the four-point correlation method. These generally show a non-conservative approximation at cycles to failure less than 105, and a conservative trend at higher values of fatigue life. The log-log scale can be misleading; the error between the prediction and approximation at 106 cycles ranges between plus and minus 60.3%. Figure 3 is a similar comparison but with the Manson-Hirschberg universal slopes approximation, Equation (1). This produces a more conservative estimation of fatigue life in the higher-cycle regime. The maximum underestimation at 1 0 6 cycles is 81.3%. However, remembering the simplicity of Equation (1), it is a much more usable approach than the four-point correlation method. The second updated universal slopes expression (Muralidharan-Manson, Equation (2)) is shown compared with the prediction (Equation (3)) for the six steels in Figure 4. This is a less conservative approximation; the maximum underestimation is 19.0% at 10 6 cycles. However, because of the better fit, the likelihood of an unsafe estimation is greater. It must be borne in mind that comparison is made with a semi-empirical model, which is itself open to variation or error. Only six materials were used for comparative purposes, and their material properties were scrutinized
Fatigue, 1994, Vol 16, July 353
Approximate strain-life fatigue crack initiation predictions: F.P. Brennan Table 1 Cyclic material properties of selected steels Brinell hardness
OUTs
try
Material
Grade
(BHN)
(MPa)
(MPa)
E (GPa)
RA %
K' (MPa)
n'
1 2 3 4 5 6
1006 1030 1035 1045 1045 4340
85 128 390 500 409
318 454 476 1343 1825 1467
224 248 270 842 1259 876
206 206 196 206 206 200
73 59 56 59 51 38
813 1545 1185 1492 2636 1950
0.21 0.29 0.24 0.09 0.12 0.13
1 0 II
......
,.
...................
, ,,,, ...................
10 e
10"1
,
(MPa) 756 902 906 1408 2165 1898
-0.13 -0.12 -0.11 -0.07 -0.08 -0.09
10 e
o o
=o
•
lOS
"~
o
-0.67 -0.42 -0.47 -0.85 -0.66 -0.64
"i._
ee_o~
o
ooB=
-
¢ ~15
1.22 0.17 0.33 1.51 0.22 0.67
,,
10 7
10 o
c
, ................................................
o
.~
c;
b
eo
o 8 o a
° o
o
10 s
o
N J
I ~:
1 0"
3E
103
10 3
10 z 10 z
.....
r
,,-
........ 1 0 =1"
103
Nf
i
--
........ 10 •
,
i ...... , 10 O
Predicted
Life
,
, ,, ...... 107
, ,,,, 10 •
(3)
,
,,,
.....................................
,.,
,
, .....
107
o° o
10 •
B¢=
10 ~
:
I
~E
: ::o
: ;
.:*
o
o
:
10 4
10-1
o
10 2 10 2
.
.
.
.
.
.
.
.
10 3
.
.
.
.
.
.
.
.
.
.
10 4 Nf
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
I 0 e --
Predicted
.
.
.
.
10 • Life
.
.
.
.
.
.
.
.
.
10 ?
.
.
.
.
.
.
1 0 •
(3)
Figure 3 Fatigue life prediction and Manson-Hirschberg universal slopes approximation
354 Fatigue, 1994, Vol 16, July
10 2
< 10 3
10" Nf
Figure 2 Fatigue life prediction and four-point correlation approximation
10 II
104i
10 n --
Predicted
101 Life
10 7
10 e
(3)
Figure 4 Fatigue life prediction and Muralidharan-Manson universal slopes approximation
before being accepted; undoubtedly, much more scatter would be seen if similar comparisons were made with a casual sample of published cyclic material data. These should be regarded as the best approximation to the prediction obtainable; i.e. anything closer than a factor of two is good. In a recent study, Ong 9 considered 49 sets of materials data and expressed predictions in terms of the percentage of data sets predicting within certain deviation ranges. In fact, these predictions are far better than Ong suggests if the material properties used satisfy Equations (12) and (13). Having looked at estimations of the prediction given by Equation (3), it is worth considering the sensitivity of the prediction to variation of the cyclic material properties. Figure 5 shows the resultant predicted fatigue lives of material 6, with a -10%, 0% and +10% error in o-'r Figures 6-8 similarly show the sensitivity of Nf to e~, b and c respectively. Again, because of the log-log scales, percentage error in Nf is tabulated in Table 2. These are shown for typical low- and high-cycle situations. Overestimation of and e~ and underestimation of b and c lead to nonconservative predictions and vice versa. In the highcycle regime, errors in c~ and c give small changes to life predictions, but overestimation of ~ and underestimation of b can lead to large overpredictions
Approximate strain-life fatigue crack initiation predictions: F.P. Brennan I 0
e -
,
,
, i i,1,i
,
,
i
......
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . .
i
.....
,o,
F
.
...
......
......
.
.....
. . . . . . . .
.
......
10 7
10 7 A r~ v
v
.!
I0 •
Bo I 0 s
10 s
÷
=z
10 4 I
104
/
I
l I 0 ~
10 a
10 2 10 =
, ,,,,
..... 10 3
, ........... 1 0 4. Nf
, ........... 101
--
Pre¢llctoc:l
,. . . . . 1 01 Lifo
* .... 10 7
* ...... 101
a
,
....
,. . . . . . . . . . . .
,
10 !
(~t)
Nf
Figure 5 Sensitivity of predicted fatigue life to 4 : - - , 0% error in 4 ; o o o +10% error in 4 ; + + + , - 1 0 % error in 4
10
............................. 10 3 10 4
101
. . . . . . . . . . . . . . . . . . . . .
,,,.
,
,
i ...... 10 s
10 s Procilcte¢l
--
Life
i
i i it . . . . 10 7
i t .t 10 I
. .
(~)
Figure 7 Sensitivity of predicted fatigue life to b: , 0 % error in b; o o o , +10% error in b; + + + , - 1 0 % error in b
......
1 0
~
.
, l l l [ T .
1
•
g ......
I
. . . .
1.11
,
. . . . . . . . . . . . . . . . . .
I
,
],,glll
I
. . . .
ITI,
•
,
. . . . . . . . . . . . . . . . . . . . .
,
. . . . .
10 ?
1 0 "r
v 10 e
o
o
10 s
10 s
l
!
10 4
I
I
10=
10 ~
g
10 z 10 z
................ 10 3
,. . . . 10 4 Nf
--
, ..... 1 0 Ii
PrOCllct.ocl
,
, ,,,,,~ 1 0 II Life
, , , ......... 107
,,, 1 01
101 I~
,
,
.......
10 3
10 4 Nf
(3)
Figure 6 Sensitivity of predicted fatigue life to e~: - - , 0% error in e~; o o o , +10% error in el,t . + + + , - 1 0 % error in eft
of fatigue life. This is worth keeping in mind, as often an average value of a selection of material constants is chosen, whereas the lower end of ~ and the upper end of the b values may be more appropriate for design purposes. It is also worth noting that the error in Nf is not proportional to errors in the constitutive material constants of Equation (3) as this expression is not linear. Finally, two common approximations for the prediction of ultimate tensile strength and fatigue strength coefficient are examined in Table 3 for the six steels. The empirical equations give reasonably good predictions of the material properties.
10 i --
PreOIcteO
10 = Lifo
10 7
,
101
(3)
Figure 8 Sensitivity of predicted fatigue life to c: - in c; o o o , +10% error in c; + + + , - 1 0 % error in c
, 0 % error
Table 2 Sensitivity of fatigue life prediction to cyclic material properties
1.1 4 0.9 4 1.1 e~ 0.9 e~
8.962 × 104 cycles
3.016 x 106 cycles
+121.5% -51,2%
+206.0% -68.7%
+6.14% -6.14%
+1.3% -1.3%
1.1 b 0.9 b
-46.9% +140.0%
-69.2% +368.0%
1.1 c 0.9 c
-32.3% +77.8%
-8.2% +22,3%
+ Overestimate; - underestimate
Fatigue, 1994, Vol 16, July 355
Approximate strain-/ife fatigue crack initiation predictions: F.P. Brennan Table 3 Estimation of ~rtrrs and Material
CrUTs = 3.45 X BHN a
Error (%)
~ = CrUTs + 345 ~
Error (%)
293 442 1346 1725 1411
-7.7 -2.6 0.2 -5.5 -3.8
663 799 821 1688 2170 1812
-2.5 -11.4 -9.4 19.9 0.2 -4.5
1 2 3 4 5 6
+ Overestimate; - underestimate; BHN Brinell hardness number ~Ref. 8
CONCLUSIONS The consistency of published cyclic material properties should be examined before adopting values for insertion into strain-life approximations. The revised universal slopes approximation of Muralidharan and Manson 2 gives the best prediction of fatigue crack initiation of the methods studied for the six selected steels in comparison with predictions based on cyclic material properties. However, this increases the likelihood of non-conservative high-cycle fatigue life predictions. It is therefore not particularly an improvement over the existing expression for design purposes. The sensitivity of predicted fatigue life to the accuracy of cyclic material data varies with the total strain range, cycles to failure and cyclic material constants. For conservative estimates of high-cycle fatigue failure, the lower value of fatigue strength coefficient and the higher value of fatigue strength exponent should be chosen from any appropriate set of cyclic material values.
NOMENCLATURE b BHN c
D E K' n p
N~ RA Ae ~e Cf ~p
Act O'a
REFERENCES 1 2 3 4 5 6 7 8 9
356
Manson, S.S. Exp. Mech. 1965, 5, 193 Muralidharan, U. and Manson, S.S. ]. Eng. Mater. Technol. 1988, 110, 55 Dowling, N.E., Brose, W.R. and Wilson, W.K. In 'Fatigue Under Complex Loading', SAE, 1977, pp. 55-84 Morrow, J. In ASTM STP 378, American Society for Testing and Materials, 1965, pp. 45-87 Raske, D.T. and Morrow, J. In ASTM STP 465, American Society for Testing and Materials, 1969, pp. 1-25 Osgood, C.C. 'Fatigue Design', Pergamon, Oxford, 1982 'ASM Handbook', 10th edn, vol 1, 1990, pp. 673-688 Boardman, B. E. In Proc. SAE Fatigue Conference', P-109, Society of Automotive Engineers, April 1982, pp. 59-73 Ong, J. H. Int. J. Fatigue 1993, 15, 213
Fatigue, 1994, Vol 16, July
OUTS
~,~s
Fatigue strength exponent Brinell hardness number Fatigue ductility exponent Ductility Modulus of elasticity (GPa) Cyclic strength or strain-hardening coefficient (MPa) Strain-hardening exponent Number of cycles to failure Percentage reduction in area Strain range Elastic strain True fracture strain Fatigue ductility coefficient Plastic strain Stress range (MPa) Stress amplitude (MPa) True fracture strength (MPa) Fatigue strength coefficient (MPa) Ultimate tensile strength (MPa) Cyclic 0.2% offset yield strength (MPa)
NDE Centre, Department of Mechanical Engineering, University College London, Torrington Place, London WCIE 7JE, UK (Received 25 November 1993) This paper describes the most commonly used approximate strain-life equations for prediction of fatigue crack initiation. Despite the development of advanced multiphase material memory models, the simple strain-life equations are useful 'first-shot' approximations of a component's resistance to fatigue crack initiation. There are, however, certain precautions that should be observed when using these equations. Selection of appropriate cyclic material properties is of utmost importance. These properties must be internally consistent; however, many values given in published literature are not. A sensitivity analysis is also presented for predicted fatigue life with respect to percentage error in cyclic material properties. (Keywords: strain Hfe; cyclic material behaviour; consistency; sensitivity; engineering design)
Engineering experience has adopted a general empirical approach to crack generation that does not consider the evolution of individual microcracks, but defines initiation in terms of bulk material properties and the local stress system. The cyclic nature of material can vary considerably from its static stress-strain behaviour. The reasons for the differing behaviour are similar to the change in material characteristics due to cold working. Materials can harden or soften in the same cyclic load sequence. The cyclic behaviour stabilizes after a number of cycles (usually quite quickly), this achievement of a convergent cyclic behaviour is known as saturation hardening. If the cyclic stress-strain behaviour of a reversed load situation is plotted directly, the result is a hysteresis loop. This hysteresis loop stabilizes after a number of cycles on realization of saturation hardening for a given strain range. Definition of the cyclic stress-strain curve can be taken as the locus of tips of the stable hysteresis loops from several tests at different completely reversed constant strain amplitudes. Knowing the cyclic behaviour of a material, it is possible to estimate degradation of the material under repeated cyclic loading. Recently there has been renewed interest in approximate empirical strain-life fatigue crack prediction methods. These first appeared 30 years ago and are still widely used as 'first shot' indications of fatigue resistance or for material optimization studies. This paper aims to remind designers of the origin of the expressions, and tries to establish the degree of accuracy that should be expected from these expressions in predicting fatigue crack initiation.
F A T I G U E LIFE FROM M A T E R I A L S D A T A The concept that cyclic life was related to plastic strain was proposed by Coffin and Manson; however, it is usually the total strain that is known rather than just the plastic component. The total strain range is a simple summation of elastic and plastic components, but unlike its constituent parts does not have a relationship to the number of cycles to failure that is a straight line on logarithmic axes. The total strain range-fatigue life curve is asymptotic to the plastic line in the low-cycle regime, and to the elastic line in the high-cycle regime (Figure 1). Manson 1 detailed two methods whereby a material's relationship of total strain range to fatigue life can be evaluated from simple tension tests. The first of these is the fourpoint correlation method. As the relationships of elastic and plastic strain components to life are
Total StrmnLine 0og) Stma
RJmge
(t)
(log) Cycles to Failure
Figure 1 Strain-life diagram 0142-1123/94/050351-06 © 1994 Butterworth-Heinemann Ltd
Fatigue, 1994, Vol 16, J u l y
351
Approximate strain-life fatigue crack initiation predictions: F.P. Brennan approximately straight lines, only two points are required to define each case. From observation of material behaviour the following points were suggested. Elastic line: (1) N f = l / 4 , (2) Nf = 105 Plastic line: (1) Nf = 10 (2) Uf = 104
e¢=2.5crf E I~ e
= 0.9 OUTS E
Ae _ ~ (2Nf)b + e; (2Nf)c 2 E
i~p ~---0.25 D 3/4 0.0132 - ee ep 1.91
= fatigue strength b = fatigue strength e'f = fatigue ductility c = fatigue ductility
(ductility) (true fracture stress)
D)
and RA is the percentage reduction in area due to waisting of the tensile specimens. By using this approximation, only the elastic modulus and two tensile properties (the ultimate tensile strength and reduction in area) are needed to predict axial fatigue life for a specified local strain range. The other approach is to assume that the slopes of the elastic and plastic lines are the same for all materials. Again, Manson 1 and Hirschberg studied the behaviour of 29 materials, including body-centred cubic, face-centred cubic, hexagonal close packed, precipitation hardened, hot and cold worked, notch ductile, notch sensitive, strain softening, strain hardening and displaying a large variation in tensile strength, elastic modulus and ductility. They presented the following equation as a reasonable representation of the total strain range relationship with cycles to failure: AI3 -- 3"5OrUT~S N f ° ' 1 2
+ O°6Nf
°'6
(1)
E The prominence of this equation in design standards and practices is due to its simplicity and application to many different types of metal. In a more recent study 2, Muralidharan and Manson, using monotonic and fatigue data for 47 materials including steels, aluminium and titanium alloys, derived a modified universal slopes equation for the estimation of fatigue life characteristics of metals. This is given as:
The first term represents the elastic strain component; the second, the plastic strain component. Knowing these material constants, the strain-life relationship can be evaluated, and then used to predict the onset of fatigue cracking. Unfortunately, these material properties are often not readily available. The following sections highlight the significance of these, their characteristics and methods for their determination. FATIGUE DUCTILITY PROPERTIES Morrow 4 proposed the form of the material properties and Equation (3), which have since been adopted as standard. Unlike the previous approximate relations where Manson et al. predicted the material cyclic behaviour from simple tensile test results, these empirical fatigue properties are established from a series of completely reversed axial fatigue tests carried to fracture on smooth specimens. From a series of fatigue tests at different strain ranges, a plot of the log of stable plastic strain amplitude against the log of fatigue life produces a series of points that closely fits a straight line. The slope of the line will vary depending on the material. The plastic strain intersection at one half cycle (one reversal) determines the level of fatigue ductility, i.e. the fatigue ductility coefficient e~. It can be approximated by 5 / ~ \ira,
)
(4)
N ( °-56
+ 1.17(~-)0"832 N f 0.09
(2)
The study showed that the influence of ductility on the elastic line is negligible. However, it is strongly influenced by the ratio of tensile strength to modulus of elasticity. Ductility is less important than indicated by the earlier universal slopes relation for plastic strain. The ductility exponent was 0.6 but is modified to 0.155. The exponent -0.12 has been revised to -0.09 for the elastic strain component, implying greater fatigue resistance as extrapolations are made to lives much greater than 106 cycles. However, before adoption of either Equation (1) or Equation (2) the particular material should be researched with regard 352
coefficient; exponent; coefficient; and exponent.
- O--ys
/(7" TS \--0.53
Ae = O.O266DO.1551~- )
(3)
where:
where
O'f ~" O'UTS(1 +
to likely fatigue behaviour. For example, the modified universal slopes equation is less accurate for AISI 4130-4340 steels below 104 cycles but more accurate than the old equation in the high-cycle regime. An important use of these relationships is to indicate the sensitivity of tensile strength and ductility on fatigue life. Ideally, relationships should be obtained for particular materials. The most common form is given as3
Fatigue, 1994, Vol 16, July
where ~r~.s is the cyclic 0.2% offset yield strength, i.e. the stress that causes 0.2% inelastic strain on the cyclic stress-strain curve. The fatigue ductility coefficient is the true strain required to cause failure in one reversal, i.e. the intersection of the line /Aep\ log~ ~ - ) - log 2Nf
(5)
at 2Nf = 1. It is related to the monotonic fracture ductility ef, which is defined as the 'true' plastic strain required to cause fracture:
I~f= ln(lO0 ZO/oRA 100 1]
(6)
Approximate strain-/ife fatigue crack initiation predictions: F.P. Brennan The true plastic fracture strain is the same value as Manson's ductility D. The slope of the plastic strain-life line may also be considered a fatigue property and is called the fatigue ductility exponent c. This is nearly a constant for metals and normally has a value between -0.5 and -0.7, with - 0 . 6 as a representative value. It is reasonably independent of chemical composition, hardness and temperature. The plastic strain-life relationship defined in terms of the ductility properties, Aep 2 - e;(2Nf)c
(7)
documented were chosen. The mechanical properties of these are shown in Table 1. The values given are taken from ref. 7, which includes many other steels; however, the six chosen were judged to be representative of a broad range of steels, and the cyclic material properties were self-consistent. Many published material properties do not exhibit self-consistency and must therefore be regarded with scepticism. The consistency can be checked by the relations b n' = -
(12)
c
is similar to the Manson-Coffin relationship except that plastic strain amplitude is used and reversals instead of cycles. FATIGUE STRENGTH PROPERTIES The first term in Equation (3) represents the elastic strain component. The fatigue strength coefficient can be related to the plastic strain component by4 acp
1'"'
2
\K']
,
= e, \ 4 1
= e; (2Ne)c
(8)
Letting n' c = b o-, = 4 (2N,) b
(9)
so
2Nf
=[trall/b \4]
(10)
The same expressions can be found by plotting the log of stress amplitude against the log of fatigue life. The fatigue strength coefficient 4 becomes the 'true' stress required to cause failure in one reversal, i.e. the intersection of the line log ~
- log 2Nf
(11)
at 2Nf = 1. It is proportional to the true fracture strength ( 4 ~ ~f) and ranges between about 690 MPa and 3400 MPa for heat-treated metals 6. The fatigue strength exponent is the slope of the line given by Equation (11). It normally is a maximum of -0.12 for softened metals and decreases to a minimum value of about -0.05 with increasing hardness. The strength properties are also referred to as 'Basquin' constants after his 'exponential law' of fatigue (1910), which is similar to Equation (9). It can be appreciated that only four of the six cyclic material constants discussed are unique; the remaining two can be derived from the four others. DISCUSSION The previous sections have set out the most established strain-life engineering fatigue crack initiation models, and have detailed various approximations and assumptions frequently employed. It is a useful exercise to study the implications of adopting approximations, especially where it is necessary to know if a fatigue life prediction is conservative, and what the approximate degree of conservatism is, or otherwise. In order to do this, six steels whose cyclic material properties are
4 K' -
(13)
A certain percentage error (up to 20%) is often tolerable, depending on the number of decimal places to which each value is reported. For example, in Table 1, n' is only given to two decimal places. Values of 0.104 and 0.095 would both be given as 0.10, but the lowest value is 8.7% less than the largest. However, in the literature it is quite common to find errors of 50-100% when the above checks are applied. Most discrepancies in material values arise from inaccurate regression or data analysis8; or too many data points are frequently considered from one test specimen, leading to adoption of data points before saturation hardening is achieved. Equations (12) and (13) are useful ways of assessing confidence in cyclic material data. Fatigue life predictions using Equation (3) were compared with the four-point correlation method and the two universal slopes approximations (Equations (1) and (2)) for the six steels. The four-point correlation method was used by employing a closedform solution devised by Ong 9. Figure 2 shows the prediction as a line having a 45 ° slope. For each material the number of cycles required to give an equal total strain range given by Equation (3) was calculated by the four-point correlation method. These generally show a non-conservative approximation at cycles to failure less than 105, and a conservative trend at higher values of fatigue life. The log-log scale can be misleading; the error between the prediction and approximation at 106 cycles ranges between plus and minus 60.3%. Figure 3 is a similar comparison but with the Manson-Hirschberg universal slopes approximation, Equation (1). This produces a more conservative estimation of fatigue life in the higher-cycle regime. The maximum underestimation at 1 0 6 cycles is 81.3%. However, remembering the simplicity of Equation (1), it is a much more usable approach than the four-point correlation method. The second updated universal slopes expression (Muralidharan-Manson, Equation (2)) is shown compared with the prediction (Equation (3)) for the six steels in Figure 4. This is a less conservative approximation; the maximum underestimation is 19.0% at 10 6 cycles. However, because of the better fit, the likelihood of an unsafe estimation is greater. It must be borne in mind that comparison is made with a semi-empirical model, which is itself open to variation or error. Only six materials were used for comparative purposes, and their material properties were scrutinized
Fatigue, 1994, Vol 16, July 353
Approximate strain-life fatigue crack initiation predictions: F.P. Brennan Table 1 Cyclic material properties of selected steels Brinell hardness
OUTs
try
Material
Grade
(BHN)
(MPa)
(MPa)
E (GPa)
RA %
K' (MPa)
n'
1 2 3 4 5 6
1006 1030 1035 1045 1045 4340
85 128 390 500 409
318 454 476 1343 1825 1467
224 248 270 842 1259 876
206 206 196 206 206 200
73 59 56 59 51 38
813 1545 1185 1492 2636 1950
0.21 0.29 0.24 0.09 0.12 0.13
1 0 II
......
,.
...................
, ,,,, ...................
10 e
10"1
,
(MPa) 756 902 906 1408 2165 1898
-0.13 -0.12 -0.11 -0.07 -0.08 -0.09
10 e
o o
=o
•
lOS
"~
o
-0.67 -0.42 -0.47 -0.85 -0.66 -0.64
"i._
ee_o~
o
ooB=
-
¢ ~15
1.22 0.17 0.33 1.51 0.22 0.67
,,
10 7
10 o
c
, ................................................
o
.~
c;
b
eo
o 8 o a
° o
o
10 s
o
N J
I ~:
1 0"
3E
103
10 3
10 z 10 z
.....
r
,,-
........ 1 0 =1"
103
Nf
i
--
........ 10 •
,
i ...... , 10 O
Predicted
Life
,
, ,, ...... 107
, ,,,, 10 •
(3)
,
,,,
.....................................
,.,
,
, .....
107
o° o
10 •
B¢=
10 ~
:
I
~E
: ::o
: ;
.:*
o
o
:
10 4
10-1
o
10 2 10 2
.
.
.
.
.
.
.
.
10 3
.
.
.
.
.
.
.
.
.
.
10 4 Nf
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
I 0 e --
Predicted
.
.
.
.
10 • Life
.
.
.
.
.
.
.
.
.
10 ?
.
.
.
.
.
.
1 0 •
(3)
Figure 3 Fatigue life prediction and Manson-Hirschberg universal slopes approximation
354 Fatigue, 1994, Vol 16, July
10 2
< 10 3
10" Nf
Figure 2 Fatigue life prediction and four-point correlation approximation
10 II
104i
10 n --
Predicted
101 Life
10 7
10 e
(3)
Figure 4 Fatigue life prediction and Muralidharan-Manson universal slopes approximation
before being accepted; undoubtedly, much more scatter would be seen if similar comparisons were made with a casual sample of published cyclic material data. These should be regarded as the best approximation to the prediction obtainable; i.e. anything closer than a factor of two is good. In a recent study, Ong 9 considered 49 sets of materials data and expressed predictions in terms of the percentage of data sets predicting within certain deviation ranges. In fact, these predictions are far better than Ong suggests if the material properties used satisfy Equations (12) and (13). Having looked at estimations of the prediction given by Equation (3), it is worth considering the sensitivity of the prediction to variation of the cyclic material properties. Figure 5 shows the resultant predicted fatigue lives of material 6, with a -10%, 0% and +10% error in o-'r Figures 6-8 similarly show the sensitivity of Nf to e~, b and c respectively. Again, because of the log-log scales, percentage error in Nf is tabulated in Table 2. These are shown for typical low- and high-cycle situations. Overestimation of and e~ and underestimation of b and c lead to nonconservative predictions and vice versa. In the highcycle regime, errors in c~ and c give small changes to life predictions, but overestimation of ~ and underestimation of b can lead to large overpredictions
Approximate strain-life fatigue crack initiation predictions: F.P. Brennan I 0
e -
,
,
, i i,1,i
,
,
i
......
i
i
. . . . . . . . . . . . . . . . . . . . . . . . . . .
i
.....
,o,
F
.
...
......
......
.
.....
. . . . . . . .
.
......
10 7
10 7 A r~ v
v
.!
I0 •
Bo I 0 s
10 s
÷
=z
10 4 I
104
/
I
l I 0 ~
10 a
10 2 10 =
, ,,,,
..... 10 3
, ........... 1 0 4. Nf
, ........... 101
--
Pre¢llctoc:l
,. . . . . 1 01 Lifo
* .... 10 7
* ...... 101
a
,
....
,. . . . . . . . . . . .
,
10 !
(~t)
Nf
Figure 5 Sensitivity of predicted fatigue life to 4 : - - , 0% error in 4 ; o o o +10% error in 4 ; + + + , - 1 0 % error in 4
10
............................. 10 3 10 4
101
. . . . . . . . . . . . . . . . . . . . .
,,,.
,
,
i ...... 10 s
10 s Procilcte¢l
--
Life
i
i i it . . . . 10 7
i t .t 10 I
. .
(~)
Figure 7 Sensitivity of predicted fatigue life to b: , 0 % error in b; o o o , +10% error in b; + + + , - 1 0 % error in b
......
1 0
~
.
, l l l [ T .
1
•
g ......
I
. . . .
1.11
,
. . . . . . . . . . . . . . . . . .
I
,
],,glll
I
. . . .
ITI,
•
,
. . . . . . . . . . . . . . . . . . . . .
,
. . . . .
10 ?
1 0 "r
v 10 e
o
o
10 s
10 s
l
!
10 4
I
I
10=
10 ~
g
10 z 10 z
................ 10 3
,. . . . 10 4 Nf
--
, ..... 1 0 Ii
PrOCllct.ocl
,
, ,,,,,~ 1 0 II Life
, , , ......... 107
,,, 1 01
101 I~
,
,
.......
10 3
10 4 Nf
(3)
Figure 6 Sensitivity of predicted fatigue life to e~: - - , 0% error in e~; o o o , +10% error in el,t . + + + , - 1 0 % error in eft
of fatigue life. This is worth keeping in mind, as often an average value of a selection of material constants is chosen, whereas the lower end of ~ and the upper end of the b values may be more appropriate for design purposes. It is also worth noting that the error in Nf is not proportional to errors in the constitutive material constants of Equation (3) as this expression is not linear. Finally, two common approximations for the prediction of ultimate tensile strength and fatigue strength coefficient are examined in Table 3 for the six steels. The empirical equations give reasonably good predictions of the material properties.
10 i --
PreOIcteO
10 = Lifo
10 7
,
101
(3)
Figure 8 Sensitivity of predicted fatigue life to c: - in c; o o o , +10% error in c; + + + , - 1 0 % error in c
, 0 % error
Table 2 Sensitivity of fatigue life prediction to cyclic material properties
1.1 4 0.9 4 1.1 e~ 0.9 e~
8.962 × 104 cycles
3.016 x 106 cycles
+121.5% -51,2%
+206.0% -68.7%
+6.14% -6.14%
+1.3% -1.3%
1.1 b 0.9 b
-46.9% +140.0%
-69.2% +368.0%
1.1 c 0.9 c
-32.3% +77.8%
-8.2% +22,3%
+ Overestimate; - underestimate
Fatigue, 1994, Vol 16, July 355
Approximate strain-/ife fatigue crack initiation predictions: F.P. Brennan Table 3 Estimation of ~rtrrs and Material
CrUTs = 3.45 X BHN a
Error (%)
~ = CrUTs + 345 ~
Error (%)
293 442 1346 1725 1411
-7.7 -2.6 0.2 -5.5 -3.8
663 799 821 1688 2170 1812
-2.5 -11.4 -9.4 19.9 0.2 -4.5
1 2 3 4 5 6
+ Overestimate; - underestimate; BHN Brinell hardness number ~Ref. 8
CONCLUSIONS The consistency of published cyclic material properties should be examined before adopting values for insertion into strain-life approximations. The revised universal slopes approximation of Muralidharan and Manson 2 gives the best prediction of fatigue crack initiation of the methods studied for the six selected steels in comparison with predictions based on cyclic material properties. However, this increases the likelihood of non-conservative high-cycle fatigue life predictions. It is therefore not particularly an improvement over the existing expression for design purposes. The sensitivity of predicted fatigue life to the accuracy of cyclic material data varies with the total strain range, cycles to failure and cyclic material constants. For conservative estimates of high-cycle fatigue failure, the lower value of fatigue strength coefficient and the higher value of fatigue strength exponent should be chosen from any appropriate set of cyclic material values.
NOMENCLATURE b BHN c
D E K' n p
N~ RA Ae ~e Cf ~p
Act O'a
REFERENCES 1 2 3 4 5 6 7 8 9
356
Manson, S.S. Exp. Mech. 1965, 5, 193 Muralidharan, U. and Manson, S.S. ]. Eng. Mater. Technol. 1988, 110, 55 Dowling, N.E., Brose, W.R. and Wilson, W.K. In 'Fatigue Under Complex Loading', SAE, 1977, pp. 55-84 Morrow, J. In ASTM STP 378, American Society for Testing and Materials, 1965, pp. 45-87 Raske, D.T. and Morrow, J. In ASTM STP 465, American Society for Testing and Materials, 1969, pp. 1-25 Osgood, C.C. 'Fatigue Design', Pergamon, Oxford, 1982 'ASM Handbook', 10th edn, vol 1, 1990, pp. 673-688 Boardman, B. E. In Proc. SAE Fatigue Conference', P-109, Society of Automotive Engineers, April 1982, pp. 59-73 Ong, J. H. Int. J. Fatigue 1993, 15, 213
Fatigue, 1994, Vol 16, July
OUTS
~,~s
Fatigue strength exponent Brinell hardness number Fatigue ductility exponent Ductility Modulus of elasticity (GPa) Cyclic strength or strain-hardening coefficient (MPa) Strain-hardening exponent Number of cycles to failure Percentage reduction in area Strain range Elastic strain True fracture strain Fatigue ductility coefficient Plastic strain Stress range (MPa) Stress amplitude (MPa) True fracture strength (MPa) Fatigue strength coefficient (MPa) Ultimate tensile strength (MPa) Cyclic 0.2% offset yield strength (MPa)