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Derivation and application of a dynamic fatigue life equation
ELSEVIER
Journal of Materials ProcessingTechnology49 (1995)279-285
Journal of Materials Processing Technology
Derivation and application of a dynamic fatigue life equation Hung-Kuk Oh*, Woong-Gi Yoon Department of Mechanical Engineering, Ajou University, Soowon 441-380, South Korea
Received 24 August 1993
Industrial Summary An Arrhenius-type static life equation is converted to a dynamic fatigue equation by integrating the fatigue damage fraction of the material. It is applied to high-cycle fatigue under fully reversed tension-compression loading. The final form of the life equation is not one of power law but one of exponential fitting. From the life equation, the fatigue potential energy Uo and the lethargy coefficient ? are acquired. Example studies are done for fatigue S-N curves for several common structural metals: cold-rolled steels, 4340 steel in the annealed condition, 7075 T-6 aluminum and 2024 T-3 aluminum.
I. Introduction By far the majority of experimental work on fatigue has been concerned with establishing the relationship between the stress amplitude and the number of cycles to failure, such work having predominantly used the fully reversed tension-compression test. Results of such experiments are generally represented on S - N diagrams, where the logarithm of the stress amplitude is plotted as a function of the logarithm of the number of cycles to failure. However, in many cases it has been stated that linear log-log fitting is more difficult than exponential fitting. Life problems are generally represented as an Arrhenius model. Zhurkov published a static fatigue life equation in 1965 [1], this equation relating temperature and static stress to life. In many cases engineering materials are loaded sinusoidal at high frequency. This static equation must therefore be converted to a dynamic type that take sinusoidal loading into consideration, in order to be able to determine the dynamic fatigue life. Finally, the number of cycles to failure is expressed as a function of temperature and alternating-stress amplitude. The characteristic values of the material's fatigue are found from the dynamic fatigue life equation, the fatigue
*Corresponding author. 0924-0136/95/$09.50 © 1995 ElsevierScience S.A. All rights reserved. SSDI 0 9 2 4 - 0 1 3 6 ( 9 4 ) 0 1 3 4 0 - 7
280
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285
potential energy, the lethargy coefficient and the alternating stress amplitude at N=I.
2. Derivation of the dynamic fatigue equation The Arrhenius model for the life equation and Zhurkov's static fatigue equation are of the same type, as follows. (1)
z = roe (v° 7a)/kT, -
where z is the life of the material, Zo is a material constant, T is the Kelvin-scale temperature, k is the Boltzman constant, Uo is the bonding energy constant of the material, y is the lethargy coefficient of the material and a is the stress applied to the material. In this static model, a and T are not functions of time. Because tr and T are functions of time in the dynamic fatigue model, the fraction of the life already expired is as follows [2]: dt ~.oe(Uo - "/a (t))/kT(t)
= fraction of the life expired in time interval dt.
(2)
The whole life is integrated as below dt
(3)
j" oe(Uo --~(t))/kT(t ) -~- 1.
3. Application of the dynamic fatigue equation to high-cycle fatigue under fully reversed tension-compression loading In the ordinary engineering fatigue test, it is assumed that the temperature is constant and that the stress is of sinusoidal variation, as Fig. 1. a(t) = ~ cos cot, T ( t ) = T = constant,
(4)
where ~ is the amplitude of the fluctuating stress. Eq. (3) becomes N(2n/w)
f
dt ,.COe ( U o - ~,a (t))/kT (t)
= 1,
z = N(2n/co),
0
where N is the number of cycles to failure. This equation is simply transformed as cot = x,
2 n f = co,
dt = dx/co.
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285
281
stress
time
Fig. 1. Loading condition.
The life equation can be integrated as
2N
~e~,~. . . . /kT dX
"CoeU°/kT ,) 0
-
(D
zoeVo/kr ~ o\~--~j = 1, failure,
(6)
where K
f
e . . . . . dx = rtlo(cZ).
(7)
0
Finally, the n u m b e r of cycles to failure is vzoeUo/kT
N -
(8)
Io(y&/kT)'
where v is the frequency, Hz. The difference between Io(x ) and its asymptote e~/(2xx) 1/2 is 0.3% for x = 0.2-500. The whole life (L) is (9)
L = Zo 2 X k T / and the n u m b e r of cycles to failure is
f,,
7~'~t/2 e tv°-r~)/kT
)
(10)
282
H.-K. Oh, I~E-G.Yoon / Journal of Materials Processing Technology49 (1995) 279-285
]
. . . . . . . . .
N~
N2
In N
Fig. 2. S-N curve.
4. Fatigue potential energy, U0; lethargy coefficient, 3'; and alternating stress amplitude at N = 1,¢~N=l Eq. (10) can be rewritten as follows:
e ~ / k T N = VZoeU°/kT( k2~t7 T) b
1/2.
(11,
Because woe v°/kr is constant and (2rc76/kT) 1/2 is slowly varying, Eq. (11) can be expressed as
eT~/kr N = Noe v°/kT = constant,
(12)
where No = 1. F r o m Eq. (12) can be defined Uo = ~'bN= 1,
(13)
where &N= 1 is the alternating-stress amplitude that gives N = 1, ordinarily bN = 1 being the same value as the ultimate tensile stress tru. T o find the lethargy coefficient, 7, Eq. (12) is used as in the following:
e~b~/kWN1 = e~/kT N2, 7(T1 + lnN1
kT
7=
762 = ~ + In N2,
k T l n N 2 -- InN1 O-1 -- bE
(14)
T h e lethargy coefficient, 7, can be found on the S - N curve relating the alternating-stress amplitude to the logarithm of the n u m b e r of cycles to failure, as Fig. 2.
5. Example studies from the S - N curves (to find the characteristic constants) Fatigue S - N curves for several c o m m o n structural metals: ((a) cold-rolled mild steel; (b) 4340 steel in the annealed condition; (c) 7075 T-6 aluminum; and (d) 2024 T-3
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285
283
aluminum); are shown in Fig. 3. In the original curves (ASTM Special Technical Publication No. 462, 1970) the logarithm of the stress amplitude is plotted as a function of the number of cycles to failure [3]. It can be seen from Fig. 3 that they are linear when presented as real plotting: this observation indicates that the relationship between the stress amplitude and the number of cycles to failure is a type of Arrhenius model.
E 70 Cr,
q~ _5
±: 50 Q
E © k
fatigue .
m
.
.
.
.
.
.
.
.
.
limit .
.
.
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.
.
.
.
.
.
.
.
.
.
.
~
(~3
cJ~ C~ r~ C ~:~ <
i
10
i IIItlll
t I LIHId
10
I ~ IIIIIII
10 2
~
I IHIIII
10 ~
, 11 II1,11 51 I , i , . , l
10 '
10
, , I1,,,11
10 ~
, ,1 , t . . l
10 '
10 B
10 g
Cycles to foilure
(a) - T 1 TTTITr~
~ I .
I
i
~ Trqrrr'r[
T T rlTrlf~
Tq
illll I ~ l
iillll
I
~
l ~ T m l l - - ~
rl
E CTI
q~ I ~q
\.\
1
3C)
\
&
\.
E
\.
0 (n
fatigue
" _ % _
limit
(3m ~j
.
.
.
.
c~ 4. ~3 c~ C q~ +,
<
(~
---'t--r~rvlTrv-~
10
(b)
, 11,,,1~--~-~ ,,,,,q
t0 ~
--1--r T'rrx'm-- ~ r - l - r r m r l T - 1- r r ~ r r r r ~ - - ~ T ~ m r ~ , . i
10 s
10'
10 ~
10'
10 '
, , ,,1,
10 B
Cyr:les [o fcJilure
Fig. 3. Fatigue S - N curves for several common structural metals: (a) cold-rolled mild steel; (b) 4340 steel in the annealed condition; (c) 7075 T-6 aluminum; (d) 2024 T-3 aluminum.
284
H.-K. Oh, W,-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285 100
~
T
-
-
, ,,,,,,,,
, ,,,,,,,, -,
,,,,,,,,
, ,rm~r--~,,,,,,,,-~,,,,,,,
,,,,,,
80 ([T,
(1) q~3
60
L-)..
£ (1 on (n (D
40
4, G']
20 f_ "g; C) C 0
10
10 2
(C)
10 ~
10'
Cycles
.._., 1 2 0 ~[_ -- ........ [
....... i
l
to
r~TTCr[
lOS
10'
10 7
'~o'
failure
i
r I lllrl I
k
i IIIIul
f
I IIIIII I
I
I i lllIl I
I
IA~LtIII
,
, IlII,IL
E 2~ ,__,
q)
,,,
80
4'
o_ r3
Q
40
+, [n
c •,L
(3
ql
11
~aaJ~l.___~a..lad.uJ._~±aAnl
I0
10 2
I0 ~
I0
Cycles to failure
(d)
Fig. 3. C o n t i n u e d
From Fig. 3, the following can be estimated. (a) Cold-rolled mild steel: 7 = kT
ln(17411 100) = 0.2079, 75.37 -- 28.12
U 0 -~- ~ 6 N =
'
1 =
0.2079 x 75.37 = 15.67,
fO ~
I0'
--I
J_u_
0 7
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279 285
285
(b) 4340 steel in the annealed condition: 7 = kT
In (358 546) = 0.0664, 152.17 - 38.69
Uo = 76N:1 = 0.0664× 152.17 = 10.1, (c) 7075 T-6 aluminum: ln(1 162980) 7 = kT
97.71 - 20.72
- 0.106882,
Uo = 73"N= 1 = 0.106882 x 92.71 = 9.91, (d) 2024 T-3 aluminum: ? = kT
In (539 604) = 0.092, 110.43 - 25.88
Uo = ? b N : I = 0.092 × 110.43 = 10.16.
6. Conclusions A dynamic fatigue equation can be derived from the Arrhenius-type static life equation by integrating the fatigue damage fraction of the material. The type of the equation is not one of power law but one of exponential fitting. The equation for fatigue potential energy and lethargy coefficient can be acquired from the dynamic fatigue equation. Example studies are done for several c o m m o n structural metals.
References [1] S.N. Zhurkov, Int. J. Fract. Mech., I, (1965) 311-323. I-2] B.H. Lee, Korea Institute of Science and Technology, Invited Lectures at Univ. of Tokyo and Univ. of Branschweig, March 1982 and July 1983. [3] N.P. Sub and A.P.L. Turner, Elements of the Mechanical Behaviour of Solids, Scripta Book Company, 1975.
Journal of Materials ProcessingTechnology49 (1995)279-285
Journal of Materials Processing Technology
Derivation and application of a dynamic fatigue life equation Hung-Kuk Oh*, Woong-Gi Yoon Department of Mechanical Engineering, Ajou University, Soowon 441-380, South Korea
Received 24 August 1993
Industrial Summary An Arrhenius-type static life equation is converted to a dynamic fatigue equation by integrating the fatigue damage fraction of the material. It is applied to high-cycle fatigue under fully reversed tension-compression loading. The final form of the life equation is not one of power law but one of exponential fitting. From the life equation, the fatigue potential energy Uo and the lethargy coefficient ? are acquired. Example studies are done for fatigue S-N curves for several common structural metals: cold-rolled steels, 4340 steel in the annealed condition, 7075 T-6 aluminum and 2024 T-3 aluminum.
I. Introduction By far the majority of experimental work on fatigue has been concerned with establishing the relationship between the stress amplitude and the number of cycles to failure, such work having predominantly used the fully reversed tension-compression test. Results of such experiments are generally represented on S - N diagrams, where the logarithm of the stress amplitude is plotted as a function of the logarithm of the number of cycles to failure. However, in many cases it has been stated that linear log-log fitting is more difficult than exponential fitting. Life problems are generally represented as an Arrhenius model. Zhurkov published a static fatigue life equation in 1965 [1], this equation relating temperature and static stress to life. In many cases engineering materials are loaded sinusoidal at high frequency. This static equation must therefore be converted to a dynamic type that take sinusoidal loading into consideration, in order to be able to determine the dynamic fatigue life. Finally, the number of cycles to failure is expressed as a function of temperature and alternating-stress amplitude. The characteristic values of the material's fatigue are found from the dynamic fatigue life equation, the fatigue
*Corresponding author. 0924-0136/95/$09.50 © 1995 ElsevierScience S.A. All rights reserved. SSDI 0 9 2 4 - 0 1 3 6 ( 9 4 ) 0 1 3 4 0 - 7
280
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285
potential energy, the lethargy coefficient and the alternating stress amplitude at N=I.
2. Derivation of the dynamic fatigue equation The Arrhenius model for the life equation and Zhurkov's static fatigue equation are of the same type, as follows. (1)
z = roe (v° 7a)/kT, -
where z is the life of the material, Zo is a material constant, T is the Kelvin-scale temperature, k is the Boltzman constant, Uo is the bonding energy constant of the material, y is the lethargy coefficient of the material and a is the stress applied to the material. In this static model, a and T are not functions of time. Because tr and T are functions of time in the dynamic fatigue model, the fraction of the life already expired is as follows [2]: dt ~.oe(Uo - "/a (t))/kT(t)
= fraction of the life expired in time interval dt.
(2)
The whole life is integrated as below dt
(3)
j" oe(Uo --~(t))/kT(t ) -~- 1.
3. Application of the dynamic fatigue equation to high-cycle fatigue under fully reversed tension-compression loading In the ordinary engineering fatigue test, it is assumed that the temperature is constant and that the stress is of sinusoidal variation, as Fig. 1. a(t) = ~ cos cot, T ( t ) = T = constant,
(4)
where ~ is the amplitude of the fluctuating stress. Eq. (3) becomes N(2n/w)
f
dt ,.COe ( U o - ~,a (t))/kT (t)
= 1,
z = N(2n/co),
0
where N is the number of cycles to failure. This equation is simply transformed as cot = x,
2 n f = co,
dt = dx/co.
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285
281
stress
time
Fig. 1. Loading condition.
The life equation can be integrated as
2N
~e~,~. . . . /kT dX
"CoeU°/kT ,) 0
-
(D
zoeVo/kr ~ o\~--~j = 1, failure,
(6)
where K
f
e . . . . . dx = rtlo(cZ).
(7)
0
Finally, the n u m b e r of cycles to failure is vzoeUo/kT
N -
(8)
Io(y&/kT)'
where v is the frequency, Hz. The difference between Io(x ) and its asymptote e~/(2xx) 1/2 is 0.3% for x = 0.2-500. The whole life (L) is (9)
L = Zo 2 X k T / and the n u m b e r of cycles to failure is
f,,
7~'~t/2 e tv°-r~)/kT
)
(10)
282
H.-K. Oh, I~E-G.Yoon / Journal of Materials Processing Technology49 (1995) 279-285
]
. . . . . . . . .
N~
N2
In N
Fig. 2. S-N curve.
4. Fatigue potential energy, U0; lethargy coefficient, 3'; and alternating stress amplitude at N = 1,¢~N=l Eq. (10) can be rewritten as follows:
e ~ / k T N = VZoeU°/kT( k2~t7 T) b
1/2.
(11,
Because woe v°/kr is constant and (2rc76/kT) 1/2 is slowly varying, Eq. (11) can be expressed as
eT~/kr N = Noe v°/kT = constant,
(12)
where No = 1. F r o m Eq. (12) can be defined Uo = ~'bN= 1,
(13)
where &N= 1 is the alternating-stress amplitude that gives N = 1, ordinarily bN = 1 being the same value as the ultimate tensile stress tru. T o find the lethargy coefficient, 7, Eq. (12) is used as in the following:
e~b~/kWN1 = e~/kT N2, 7(T1 + lnN1
kT
7=
762 = ~ + In N2,
k T l n N 2 -- InN1 O-1 -- bE
(14)
T h e lethargy coefficient, 7, can be found on the S - N curve relating the alternating-stress amplitude to the logarithm of the n u m b e r of cycles to failure, as Fig. 2.
5. Example studies from the S - N curves (to find the characteristic constants) Fatigue S - N curves for several c o m m o n structural metals: ((a) cold-rolled mild steel; (b) 4340 steel in the annealed condition; (c) 7075 T-6 aluminum; and (d) 2024 T-3
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285
283
aluminum); are shown in Fig. 3. In the original curves (ASTM Special Technical Publication No. 462, 1970) the logarithm of the stress amplitude is plotted as a function of the number of cycles to failure [3]. It can be seen from Fig. 3 that they are linear when presented as real plotting: this observation indicates that the relationship between the stress amplitude and the number of cycles to failure is a type of Arrhenius model.
E 70 Cr,
q~ _5
±: 50 Q
E © k
fatigue .
m
.
.
.
.
.
.
.
.
.
limit .
.
.
~3.o .
.
.
.
.
.
.
.
.
.
.
.
~
(~3
cJ~ C~ r~ C ~:~ <
i
10
i IIItlll
t I LIHId
10
I ~ IIIIIII
10 2
~
I IHIIII
10 ~
, 11 II1,11 51 I , i , . , l
10 '
10
, , I1,,,11
10 ~
, ,1 , t . . l
10 '
10 B
10 g
Cycles to foilure
(a) - T 1 TTTITr~
~ I .
I
i
~ Trqrrr'r[
T T rlTrlf~
Tq
illll I ~ l
iillll
I
~
l ~ T m l l - - ~
rl
E CTI
q~ I ~q
\.\
1
3C)
\
&
\.
E
\.
0 (n
fatigue
" _ % _
limit
(3m ~j
.
.
.
.
c~ 4. ~3 c~ C q~ +,
<
(~
---'t--r~rvlTrv-~
10
(b)
, 11,,,1~--~-~ ,,,,,q
t0 ~
--1--r T'rrx'm-- ~ r - l - r r m r l T - 1- r r ~ r r r r ~ - - ~ T ~ m r ~ , . i
10 s
10'
10 ~
10'
10 '
, , ,,1,
10 B
Cyr:les [o fcJilure
Fig. 3. Fatigue S - N curves for several common structural metals: (a) cold-rolled mild steel; (b) 4340 steel in the annealed condition; (c) 7075 T-6 aluminum; (d) 2024 T-3 aluminum.
284
H.-K. Oh, W,-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279-285 100
~
T
-
-
, ,,,,,,,,
, ,,,,,,,, -,
,,,,,,,,
, ,rm~r--~,,,,,,,,-~,,,,,,,
,,,,,,
80 ([T,
(1) q~3
60
L-)..
£ (1 on (n (D
40
4, G']
20 f_ "g; C) C 0
10
10 2
(C)
10 ~
10'
Cycles
.._., 1 2 0 ~[_ -- ........ [
....... i
l
to
r~TTCr[
lOS
10'
10 7
'~o'
failure
i
r I lllrl I
k
i IIIIul
f
I IIIIII I
I
I i lllIl I
I
IA~LtIII
,
, IlII,IL
E 2~ ,__,
q)
,,,
80
4'
o_ r3
Q
40
+, [n
c •,L
(3
ql
11
~aaJ~l.___~a..lad.uJ._~±aAnl
I0
10 2
I0 ~
I0
Cycles to failure
(d)
Fig. 3. C o n t i n u e d
From Fig. 3, the following can be estimated. (a) Cold-rolled mild steel: 7 = kT
ln(17411 100) = 0.2079, 75.37 -- 28.12
U 0 -~- ~ 6 N =
'
1 =
0.2079 x 75.37 = 15.67,
fO ~
I0'
--I
J_u_
0 7
H.-K. Oh, W.-G. Yoon / Journal of Materials Processing Technology 49 (1995) 279 285
285
(b) 4340 steel in the annealed condition: 7 = kT
In (358 546) = 0.0664, 152.17 - 38.69
Uo = 76N:1 = 0.0664× 152.17 = 10.1, (c) 7075 T-6 aluminum: ln(1 162980) 7 = kT
97.71 - 20.72
- 0.106882,
Uo = 73"N= 1 = 0.106882 x 92.71 = 9.91, (d) 2024 T-3 aluminum: ? = kT
In (539 604) = 0.092, 110.43 - 25.88
Uo = ? b N : I = 0.092 × 110.43 = 10.16.
6. Conclusions A dynamic fatigue equation can be derived from the Arrhenius-type static life equation by integrating the fatigue damage fraction of the material. The type of the equation is not one of power law but one of exponential fitting. The equation for fatigue potential energy and lethargy coefficient can be acquired from the dynamic fatigue equation. Example studies are done for several c o m m o n structural metals.
References [1] S.N. Zhurkov, Int. J. Fract. Mech., I, (1965) 311-323. I-2] B.H. Lee, Korea Institute of Science and Technology, Invited Lectures at Univ. of Tokyo and Univ. of Branschweig, March 1982 and July 1983. [3] N.P. Sub and A.P.L. Turner, Elements of the Mechanical Behaviour of Solids, Scripta Book Company, 1975.