Opening and Mixed Mode Fatigue Crack Growth Simulation
Under Random Loading
P. C. Gope1, S. P. Sharma2, B. Kumar2
Mechanical Engineering Department, College of Technology,
G. B. Pant University of Agriculture & Technology, Pantnagar-263145, India
E MAIL: pcgope@rediffmail.com
2.
Mechanical Engineering Department, National Institute of Technology,
Jamshedpur-831014, India
1
ABSTRACT
A model for the crack growth analysis under mixed mode random loading that includes the loading sequence effect is presented. The model
was applied for the analysis of the crack growth life under different random loading on the sheets of two different aluminum alloys, 2024T3 and 2199-T851. Analyses were carried out on the simulated loading histories obtained from different spectral densities of the nominal
stress and various loading history lengths for each case.
INTRODUCTION
Engineering structure and component contains crack of varying size, shape, and orientation. The randomly oriented crack produces mixed
mode loading. In the mixed mode loading condition a crack will propagate in a non-self similar manner. Fracture mechanics under mixed
mode loading condition as a part of the failure analysis can be studied in different steps out of which the study of crack growth direction
and crack propagation rate are most important. Several criteria have been proposed so far for the prediction of the crack growth direction
under mixed mode condition. Some of these are, maximum tangential stress criterion (MTS), strain energy density criterion (SED), or
Griffith-Irwin Energy criterion [1-3].
Several parameters have been suggested by different authors to correlate the fatigue crack growth rate under mixed mode condition. Most
of these are inform of Paris Equation. The parameters which have been used to correlate fatigue crack growth rate under mixed mode
condition are effective Stress Intensity Factor, SED factor, J-integral, Equivalent strain intensity factor etc. A good correlation of mixed
mode crack growth rate data have been seen by some of the aforementioned parameters, but there is no single parameter which gives
satisfactory correlation under all loading conditions.
The load non-proportionality, over load, crack closure and T-term has influences on the fatigue crack growth behavior under mixed mode
loading. Extensive studies of the effect of these factors on mixed mode crack growth rate have not been conducted yet now. Only a few
studies on crack growth characteristics are available in the literature. A few of them are the study of Nayes-Hashemi, Hwang et.al [4] and,
Nayes-Hashemi [5] in which they studied the effect of the mode II over load on subsequent mode I crack growth using four point and three
point bending and shear specimens made of AISI 4340 steel. The number of studies of the crack growth dealing with the factor like over
load, and T-stress term is very limited, and the roles and contributions of these factors in mixed mode crack growth are far from clear yet.
In this paper the effect of the over load on the crack growth characteristics has been studied. The different type of random loading has been
considered for mixed mode fatigue crack growth studies. In the present paper different random loading pattern has been generated and used
in the crack growth studies. The simulated results are also compared with the available experimental results.
SPECTRUM LOADING GENERATION
There are two basic ways of obtaining random loading histories, namely (a) directly from the system under service condition (b) by
computerized simulation. In the present study all the load history used in the crack growth analysis are simulated through computer.
Different loading histories defined by their spectral density function and length (represented by number of the peaks) were generated. Six
different form of the spectral density function of normal stresses, S(w) have been used. Table 1 lists the characteristic values of the spectral
densities considered. The meaning of each symbols are illustrated in Fig. 1
Table 1. Parameters of the power spectral densities, all have the same mean square.
W2
W3
W4
H1/H2
TyW1
α*
pe
H1
D
18
20
0.9982
0.85
E
7
27
H2
F
5
40
0.795
G
5
15
80
130
2.0
0.64
60
2.0
0.705
H
5
25
30
P
5
15
25
75
6.67
0.756
W1
W2 W3
W4
Fig. 1 Spectral densities
α * is a parameter descriptive of the load irregularities and indicative of band width. It is defined as
α* =
M2
(1)
M 0M 4
∞
where M n = ∫ wn S ( w)dw
(2)
−∞
is the nth order moment. Six loading history length, L of 300, 500, 1000, 2000, 3000 and 5000 peaks were taken. For each
type of spectrum density and number of peaks, 25 loading histories were generated. The simulation was done by
superimposing the sinusoidal function of the random phase with uniform distribution. The mean square stress of each loading
histories were kept at constant value of 1947 MPa 2. Each group were designated by letter L followed by number of peaks
and type of spectra. Example L300G means G type of loading spectra with 300 numbers of peaks. Some of the results are
shown in Fig. 2.
160
60
Stres s Amplitude, M Pa
Powe r Spec trum M agnitude (dB)
140
40
20
0
120
100
80
60
40
20
0
-20
-40
0
0.2
0.4
0.6
Frequency
0.8
0
100
200
300
400
H is t o r y le n g t h
500
(c)
Fig. 2 (a) Simulated PSD (b) Assumed PSD
Generated Load history for bandwidth of 0.795
1
(a)
600
(c)
(b)
5
40
w
The power spectral density for other random loading types E-P described in Table 1 are shown in Fig. 3. Fig. 4 shows
simulated load histories for different spectral density function.
60
60
(b ) L 1 0 0 0E
(a ) L 10 00F
P o we r S p e c trum Mag nitud e (d B )
P o we r S p e c trum Mag nitud e (d B )
50
40
30
20
10
0
-1 0
-2 0
-3 0
0
0 .2
0 .4
0 .6
F re q ue nc y
0 .8
1
40
20
0
-2 0
-4 0
-6 0
0
0 .2
0 .4
0 .6
F re q ue nc y
0 .8
1
80
60
(f) L100 0D
P ow e r S p ec tru m M a g n itu d e (dB )
P o w er Sp ectrum M ag nitud e (dB )
(c) L1000G
50
40
30
20
60
40
20
0
-20
-40
-60
-80
10
0
0.2
0.4
0.6
Frequency
0.8
(d) L1000H
50
180
40
160
30
140
20
120
10
0.8
1
0
100
80
60
-10
40
0
0.2
0.4
0.6
Frequency
0.8
1
20
0
200
60
(e) L1000P
400
600
S tre s s C y c le s
800
1000
800
1000
180
(b ) L o a d h is to ry fo r L 1 0 0E
50
160
140
40
S tres s, M P a
P o w er S p ec tru m M ag n itu d e (d B )
0.4
0 .6
Frequency
(a ) L o a d h isto ry fo r L 10 0 P
S tre ss , M P a
P o w er S p ec tru m M ag n itu d e (d B )
0.2
Fig. 3 Simulated PSD for assumed spectral density
function (Refer Table 1)
60
-20
0
1
30
20
120
100
80
60
40
10
0
0.2
0.4
0.6
Frequency
0.8
1
20
0
200
400
600
S tre ss c y cles
Fig. 4 Simulated load histories for different spectral
density functions
CRACK GROWTH MODEL
Most of the crack growth rate equations proposed for mixed mode loading are based on either strain energy density approach or stress
intensity factor approach. Sih & Bathelemy [6] have proposed the following relation
da
= 0.02(∆S )1.58
(3)
dN
where da/dN is in m/cycle and ∆S is in MJ/m2.
Patel & Pandey [7] proposed the following relation to study the crack growth rate
n/2
da
⎛ 4πµ
⎞
= C⎜
∆S ⎟
(4)
dN
⎝ 1 − 2υ
⎠
where C and n are mode I Paris constants. The comparison of the above two equations was made with the experimental data of 2024-T3
aluminum alloy. It is found that the experimental data points are not in agreement with the predicted values obtained from Eqns. 3 and 4 for
given ∆S . Equation given by Sih & Bathelemy predicts the higher crack growth rate where as Eqn. 4 predicts lower rate as compared to
the experimental crack growth data [10].
The crack opening stress is not included in any one of the above SED based models. Hence, a closure parameter based on strain energy
density factor has been introduced in the present crack growth model. From the experimental results, the following type of equation has
been obtained.
da
= (8.2186) 10 − 07 (∆Us )1.0394
dN
(5)
where da/dN is in m/cycle and
∆S min − ∆Sop
∆Us =
∆Scri − ∆Sop
(6)
where ∆S min = S max − S min
min
min
(7)
max = S (θ , σ
min
and S min
0 max ) , is the maximum strain energy density factor , Smin = S (θ0 ,σ min ) , is the minimum strain energy density
factor, ∆Sop = S (θ0 ,σ op ) , is the opening strain energy density factor, ∆S cri = S (θ 0 , σ cri ) , is the critical strain energy density factor. S
is the strain energy density factor and given elsewhere [6]
CRACK CLOSURE MODEL
Newman [8] calculated the crack opening stress for CCT specimen subjected to uniaxial constant amplitude loading, and obtained the
following equation fitted to the numerical results.
SOP
= A0 + A1R + A2 R 2 + A3R3 , R ≥ 0
(8)
Smax
SOP
= A0 + A1R , −1 ≤ R < 0
(9)
Smax
SOP is the opening stress, Smax is the maximum stress. The coefficients are given as,
1/ α
⎡ ⎛ π Smax ⎞⎤
S
⎟⎥
A0 = (0.825 − 0.3α + 0.05α 2 ) ⎢cos⎜⎜
, A1 = (0.415 − 0.071α ) max , A2 = 1 − A0 − A1 − A3 , A3 = 2 A0 + A1 − 1
⎟
2
σ
σ0
0 ⎠⎥⎦
⎣⎢ ⎝
The above equation is a function of stress ratio, R, stress level, Smax, and three dimensional constraint factor
plane strain conditions are simulated with
equation for
α c . Generally, plane stress or
α c =1 or 3, respectively. Recently McMaster and Smith [9] presented a third order polynomial
α c in the range of 0.27 < α c < 1.15
for 2024-T351 Al alloy. The variation of constraint factor as given by them is:
⎧1 .0 if φ > 1 .15
⎪
⎪
αc = ⎨
3 . 0 if φ ≤ 0 .27
⎪
⎪ − 0 .754 φ 3 + 5 . 179 φ 2 − 8 .219 φ + 4 .864 ,0 .27 < φ < 1 .15
⎩
φ = ∆K /(σ 0 t ) , t is the specimen thickness, σ 0 =
Incorporating this in Eqn. 8 we can derive as [10-11]
SOP
= 0.3725 + 0.6275( A0 + A1R + A2 R 2 + A3R3 )
Smax
(10)
σ y + σU
2
(11)
P rersent model
M odel (derived from S trip-yield model)
M odel (derived from strip-yield model)
2024-T 3, 60 deg crcak angle
2024-T 3, 45 deg crack angle
2024-T 3, 30 deg crack angle
1
0.95
0.9
S o p /S m ax
0.85
P lane stress
0.8
0.75
P lane strain
0.7
0.65
0.6
0.55
0
0.2
0.4
0.6
0.8
1
R
Fig. 5 Comparison of predictions of opening stress with extended Newman’ equation and experimental data
The variation of above equation for different constraint factors is shown in Fig. 5 along with some of the experimental results. Figure
shows that Sop/Smax can be well approximated by properly selecting the constraint factor.
MATERIALS & METHOD
The materials on which the simulations were carried out is 2024-T3 Al alloy. The material properties are E=67850.0 MPa, σ y = 370MPa ,
υ = 0.334 , KI, th=2.75 MPa m , ∆ KIC=50.22 MPa m .Cycle-by-cycle simulation of crack growth from specified crack length aI=
10 mm to final crack length af = 25 mm has been carried out. The growth life times obtained for different loading history and different
groups are presented in Figs 7-11. The procedure is explained in Fig. 6. The specimen geometry is described in reference [10-11]
RESULTS & DISCUSSION
Mode I Fatigue life under random loading
To prove the validity of the presented model, another group of random loads (M81,...) available in literature [12] are considered. The
growth lifetimes were predicted using the presented model. Table 3 shows the summary of the predicted ratio and for different loading
along with the predicted results given in reference (ASTM STP 748). The average ratio and standard deviation estimated from the results of
present method are 1.118 and 0.259 where as these values are 0.846 and 0.213 obtained from JC(1) model. JC(1) is based on Walker Crack
growth rate equation, compressive stress set to zero, and tensile load interaction not accounted for.
Table 3. Absolute comparison of the test and predicted life of 2219-T851 Al alloy under mode I condition
Load
M81
M82
M83
M84
M85
M88
M89
M90
M91
M92
aI
(mm)
4.06
3,81
3.81
4.00
3.87
3.81
3.81
3.81
3.81
3.81
af
(mm)
13.0
35.4
23.3
55.9
32.8
45.8
38.4
51.6
36.1
29.5
Test
Present
115700
58585
18612
268908
36397
380443
164738
218151
65627
22187
179875
59957
15000
329850
35314
357101
184149
331834
78269
18315
Life
ratio
1.554
1.023
0.806
1.227
0.970
0.939
1.118
1.521
1.193
0.826
Mixed Mode Fatigue life under Random loading
Simulations of crack growth are carried out considering the generated load histories described above. The fatigue lives obtained are
presented below in Figs. 7-11.
Input initial, final crack length, loading history
and constants of crack growth model used in the
analysis, crack angle etc
H
Enter over load factor, XOL
Calculate over load SOL= XOL. Smax
Initialization icycle=1
I=1
Jol=0
G
F
I=I
Calculate stress intensity factors,
stress ratio
Compute opening stress Sop(I) from emperical relation
Sigop=Sop(I)
N
No
Is
Smax(I) ≥ SOL
Yes
Y
N
Y
Iol=I
Is
Iol < 1
Yes
Sigop=Sop(I)
No
No
Search naxt maximum stress which is
greater than SOL. Find position j where
Smax ≥ SOL
Is
Sigop ≥ Smax
da=0
B
Evaluate Stress
Intensity Factors
Jol=j
Yes
D
Find rms of Smax & Smin between two
consecutive overloads Smax(Iol) &
Smax(Jol).Rms values are Sopx1 & Sopx2
Evaluate opening stress intensity factors
KIop< KImin
No
Iol < I
Yes
Yes
Compute crack opening stress taking maximum rms
Sop( I ) = Sopx +
Yes
Sop(I)=Sopx
Jol − I − 1
( Sop( Iol ) − Sopx )
Jol − Iol − 1
Sop(I) ≤ 0
KIop= KImin
Compute crack extension angle
Compute ∆S min
Compute ∆S Cri , ∆S OP
Compute critical condition
No
R
Sigop=Sop(I)
B
R
Yes
No
∆S min ≥ ∆S Cri
Compute Crack increment using Growth model
Compute effective crack angle, Crack length
Component fails
Write Life
Compute plastic zone size, rp
Stop
No
Yes
Smax ≤ SOL
arnew = a + rp
arol = a + rp
No
Yes
I=Iol
No
arnew > arol
No
Yes
Iol < 1
No
Yes
a ≥ af
Yes
Write a, icycle
I=I+1
Ixycle=icycle+1
D
Change XOL
Stop
I = n max
No
H
G
Yes
F
Fig. 6 Flow chart for the crack growth studies including sequence effect
Probability distribution: Fig. 7(a-b) shows the probability distribution of fatigue life for α = 600 for bandwidth, α * =0.756 and history
length of 1000 peaks of stress on normal and Weibull probability paper. It is seen that normal distribution is well fitted to the data than
Weibull distribution. Fig. 8 shows the effect of the bandwidth on the probability distribution of the life. The probability distribution of two
different type of random loading obtained for bandwidth 0.795 and 0.705 and same history length of 1000 stress peaks are shown for crack
position α = 300 on normal probability paper. From the figures it can be seen that in case of life obtained from the high bandwidth the fit is
excellent on normal and Weibull probability paper whereas for low bandwidth normal distribution can be best approximated. It is clear
from the figure 8 that all bandwidths except the narrowest bandwidth α * =0.9982 can be grouped into one category whereas life data with
bandwidth 0.9982 shows different behavior and shows almost constant life for all 25 tests although the loading groups are generated
randomly. This may be due to extreme narrow bandwidth where sequence effect can be considered as small due to quick repetition of the
load causing the sequence effect. This loading history shows similar effect as that of constant amplitude loading. Fig. 9 shows the effect of
history length on the probability distribution of the fatigue life. The presented results show that, between the two distributions, the normal
distribution produces a better overall fit to the fatigue life data. The similar results are presented by Domigenez et al [12] for horizontal
crack (mode I) under random loading.
Mean life, µ : The effect of bandwidth and history length on mean fatigue life is shown in Fig. 10 for α = 300. It is seen that mean life
remains constant for random loads having history lengths more than 1000 peaks. The history length less than 1000 shows major influence
in the mean life. The bandwidth has also major influence on the mean fatigue life. It is seen that mean life increases with decrease in
bandwidth up to certain bandwidth approximately equal to the bandwidth of white noise (0.795), there after, it decreases. These types of
variations have been observed for all history length.
0.997
0.99
100
(a)
(a)
0.98
80
0.95
P ro b a b ility, %
Probability
0.90
0.75
0.50
0.25
0.10
0.05
0.02
0.01
60
=0.9982
0.85
0.795
0.64
0.705
0.756
40
20
0.003
0.001
1.5
1.7
1.9
2.1
Life, cycles
2.3
2.5
x 10
0.96
(b)
2
2.25 2.5 2.75
Life, Cycles
3
3.25
3.5
x 010
5
100
B a n d w id th = 0.64
0.90
c ra c k a n g le =3 0 d e g
80
0.75
0.50
P ro b ab ility, %
Probability
1.75
Fig. 8 Probability distribution of Fatigue life for α = 30
0.999
0.99
0
1.5
5
0.25
0.10
0.05
0.02
60
L3
L10
L20
L30
L50
40
0.01
20
0.003
0.001
1.6
1.8
2
2.2
Life, cycles
2.4
x 10
Fig. 7 Probability distribution of fatigue life for α = 60 0 ,
L100P type load on (a) Normal probability paper (b) Weibull
probability paper
5
0
1.5
2
2 .5
L ife, cy cles
3
3 .5
x 10
Fig. 9 Probability distribution of fatigue life for different HL
5
x 10
5
= 0.998 2
0.850
0.795
0.640
0.705
0.756
4
M ea n life , cy cles
3.5
3.5
3
x 10
5
L300
L1000
L3000
3
Life, cycles
4.5
2.5
2
(a) Crack angle = 30 deg
2.5
2
1.5
(a) cra ck an g le = 3 0 d eg
1
0
10 00
200 0
3 00 0
4 000
H isto ry len g th , L
5 000
60 00
1.5
125
135
145
155
Highest peak stress, MPa
165
Fig 10 Effect of band width on mean fatigue life
Fig 11(a) Effect of highest peak stress on fatigue life
3.5
x 10
= 0.850
0.795
0.756
0.705
0.640
3
Life, cycles
5
2.5
2
(b) Crack angle = 30 deg
1.5
120
130
140
150
160
170
Highest peak stress, MPa
180
190
Fig 11(b) Effect of highest peak stress on fatigue life
The effect of highest peak stress generally known as over load in the load history on fatigue life is shown in Fig. 11 for different
bandwidths and history lengths. It is seen that as the peak stress increases the resulting mean fatigue life alos increases. This is due to
retardation mechanism occurring due to overload.
CONCLUSIONS
In the present work a crack growth model derived considering experimental crack opening and growth results of aluminum alloys was
used. The results reported here are concluded as follows:
1. The history length to be used is a decisive element in making a test or simulation of crack growth process under random loading.
2. The use of too short histories and their repetition until failure may produce non conservative results and provide much longer lives than
those obtained with longer loading histories, which will be closer to facts.
3. In estimating fatigue lives under random loading it is required to generate or test with different representative random loading histories
and to determine the life distribution.
4. Life obtained for bandwidth between 0.705 to 0.795 shows excellent fit both on Normal and Weibull Probability paper. Narrowest or
broadest bandwidth shows poor fit on Weibull probability paper
5. Peak stresses in the loading cycles has significant effect on opening stress and hence on the crack growth.
6. Mean life increases with decrease in Bandwidth up to certain Bandwidth equals to Bandwidth of white noise, there after it decreases
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