Specimen Geometry and Material Property Uncertainty Model for
Probabilistic Fatigue Life Predictions
Prakash Chandra Gope*, Sandeep Bhatt**, and Mohit Pant**
*Associate Professor, Mechanical Engineering Department, College of Technology, G B Pant
University of Agriculture & Technology, Pantnagar-263145, India
**Research Scholar
E mail: [email protected] , Phone +91941159916, Fax +915944233338
ABSTRACT
The present paper describes an analytical model derived from energy theorem which is coupled with the uncertainty
associated with material properties and geometrical parameter to estimate fatigue life and crack growth at different level of
probability and confidence level. A substantial amount of published data has been used to validate the proposed model for
predicting S-N curve as well as fatigue crack growth of steel, aluminum alloys, copper alloys and titanium alloys. Some of
are 4340, 304 steel, 0.26 % Carbon steel, different aluminum alloys, titanium alloys etc. Different test geometry such as
CT specimen, SEN specimen geometry etc has also been taken to find out the acceptability of the model.
INTRODUCTION
The analysis of cracks within structure is an important aspect if the damage tolerance and durability of structures and
components are to be predicted. As part of the engineering design process, engineers have to assess not only how well
the design satisfies the performance requirements but also how durable the product will be over its life cycle. Often cracks
cannot be avoided in structures; however the fatigue life of the structure depends on the location and size of these cracks.
In order to predict the fatigue life for any component, crack growth study needs to be performed.
Fatigue life is related to and is affected to a great extent with the uncertainties in both the material properties and the
specimen or component geometrical parameters. The intent of the work is to contribute to the fundamental understanding
of fatigue life and its relation with these uncertainties. Fatigue life data exhibits wide scattered results due to inherent
microstructural inhomogeneity in the material properties even if the test specimens are taken from the same lot and tested
under same loading condition. Several methods [1-27] have been discussed in the literature to predict the fatigue life but
the problem of uncertainty associated with material and specimen geometry has not been addressed so far. As the fatigue
testing is time consuming and costly, setting up of an analytical method for prediction of fatigue life is necessary.
The different monotonic material properties such as yield strength, modulus of elasticity, fracture properties such as critical
stress intensity factor, threshold and crack opening stress intensity factors, loading parameters and geometry parameters
such as specimen or component width, thickness, length , crack aspect ratio etc. have been considered in the model.
These parameters are treated as random variable and assuming normally distributed, the associated uncertainty in these
parameters are incorporated in the crack growth model. The assumptions of these variables are also verified from the
simulated results. Crack growth model is derived from the energy theorem. In the present work an approximate analytical
model derived from the energy theorem and probabilistic nature of material properties and specimen geometry parameters
are combined and correlated to determine the associated error in the predicted fatigue life. The Error in estimating the
fatigue life is derived from the expected values of fatigue life. The prediction is based on minimization of the error. The
model can also be used to predict the fatigue life or crack growth at different level of probability and confidence level.
Hence it can be used to draw P-S-N curve.
MATERIAL AND METHOD
The fatigue crack propagation rate equation based on strain energy release rate derived by Bhatt [2] is given as
(
)
0.139
da
=
µ∆K I − µ∆Kth 2
dN αµ E .µσ y
and fatigue life is obtained as,
1
⎛ K
1 − ⎜ max
⎜ µ∆K
IC
⎝
⎞
⎟
⎟
⎠
2
(1)
af
N f = ∫
a0
da
0 . 139
µ ∆ K I − µ ∆ K th 2
αµ E .µ σ y
(
where
)
1.0
α=
φ
(2)
1
⎛ K
max
1− ⎜
⎜ µ ∆K
IC
⎝
⎞
⎟
⎟
⎠
2
if φ >1.15
if φ ≤ 0.27
3
-0.75 φ + 5.179φ 2 − 8.219φ + 4.864
∆K
3.0
=
for
0.27< φ <1.15
σy B
B is the specimen thickness, and ∆K I = K max − K min if Kmin > Kop otherwise ∆K I = K max − K op
where
s
: Standard deviation
da
: Increment in crack length, mm
dN
: Increment in number of cycles to failure, cycles
a0
af
: Final crack length, mm
∆K I
: Stress intensity factor range, MPa m
Nf
: Fatigue life, cycles
: Initial crack length, mm
µ
σy
µ(......)
: Population mean of variable (…..)
s(.....)
: Standard deviation of variable (…..)
K max
: Maximum stress intensity factor, MPa m
K min
: Minimum stress intensity factor, MPa m
: Population mean
: Yield strength, MPa
K op
: Crack opening stress intensity factor, MPa m
The opening parameters can be obtained from the following relations.
S op
= A0 + A1 + A2 R 2 + A3 R 3 , R ≥ 0
S max
(3)
S op
= A0 + A1 R , −1 < R < 0
S max
(4)
where
1
⎡ ⎛ π S max ⎞⎤ α
2
⎟⎥
A0 = (0.825 − 0.3α + 0.05α ) ⎢cos⎜⎜
⎟
⎣⎢ ⎝ 2 σ 0 ⎠⎥⎦
S
A1 = (0.415 − 0.017α ) max
σ0
A2 = 1 − A0 − A1 − A3
A3 = 2 A0 + A1 − 1
sop = Opening stress, Smax = Maximum stress level
The fracture is taken as the condition at which K max ≥ K IC .The N f so obtained from the solution of Equation (2)
depends on several monotonic strength parameters like modulus of elasticity
(E ) ,
( )
yield strength σ y , and fracture
parameters like threshold stress intensity factor (K th ) , critical stress intensity factor (K IC ) .It is well established that all
such strength parameters are random variable and follows normal distribution. Due to the scattered nature of these
variables fatigue life obtained can also be assumed as a random variable. The error in estimating the fatigue life can be
obtained from [8-10]
⎛1
⎞
R = ±φ N .t. ⎜ +u p 2 ⎛⎜ψ 2 − 1⎞⎟ ⎟
⎝
⎠⎠
⎝n
(5)
s
where φ N =
X + u pψ ⋅ s
and
⎛ n −1⎞
Γ⎜
⎟
⎛ n −1⎞ ⎝ 2 ⎠
ψ= ⎜
⎟
⎝ 2 ⎠ Γ⎛ n ⎞
⎜ ⎟
⎝2⎠
where µ and σ are the population mean and population standard deviation, respectively, and
u p is the normal deviate
corresponding to the probability of survival p and n is the sample size. In this study the scattered behavior of fatigue life
and crack growth is incorporated by randomizing the various fatigue parameters considering them as statistically
independent random variables following normal distribution with mean values X and standard deviation s . Equations 1-5
have been solved to predict the fatigue life. The population mean (µ X ) used in the solution can be obtained from
µ X = X + K .s
(6)
K is a factor whose value varies between -3 to 3 for normal distribution. In the present investigation K is randomly
generated. The fatigue life corresponds to the life with minimum error obtained from the equation 5. All computations are
done using MATLAB 7.0. The detailed procedure is explained in reference [2]. Brief procedure is presented below.
Algorithm
1.
2.
3.
4.
5.
6.
7.
8.
Estimate the statistical mean of model parameters from the experimental mean and standard deviation. For
computing these generate random number K (Equation (6)). The limiting values of K are ± 3.
Select type of specimen geometry (SEN, CT, CCT etc) and estimate statistical mean values of geometry
parameter similar to step 1.
If load history is not given generate the load history. The procedure is explained in reference [8-10].
Compute crack opening parameters.
Solve Equation (1) or (2) and estimate fatigue life or crack length.
Repeat steps 1 to 5 if sample size (number of predicted life) is less than 3.
Estimate Error from Equation (5) for a given probability and confidence level.
If Error determined at step 7 is less than pre assumed tolerable error, write fatigue life as the mean value of all
lives determined and corresponding stress level, otherwise go to step 1.
The material properties used in the fatigue life simulation are presented in Table 1.
Table 1. Mechanical & geometrical parameters of material used in fatigue life simulation
Material
7075-T6 al
alloy
D16 AT Al
2024 T3
4340 steel
304 Steel
0.26C
steel
6061 T6
Parameters
Mean values
Co-eff. of var (%)
Mean values
Co-eff. of var (%)
Mean values
Co-eff. of var (%)
Mean values
Co-eff. of var (%)
Mean values
Co-eff. of var (%)
Mean values
Co-eff. of var (%)
Mean values
Co-eff. of var (%)
Modulus
( GPa)
68.9
6.8
64.27
5.62
68.9
5.62
207
5.74
207
5.24
201
4.9
68.67
6.00
YS
(MPa)
540
5.8
296
5.68
393
6.00
862
5.77
276
5.98
246
4.98
295
5.95
Aspect
ratio
K IC
K th
K op
MPa m
MPa m
MPa m
*
0.2
5.3
3.2
4.2
0.2
3.2
3.2
2.19
0.2
3.0
3.2
*
0.2
*
3.2
*
0.2
*
3.2
*
0.2
*
3.2
*
0.2
*
3.2
* simulated results
30
5.9
31
5.55
5.5
5.81
99
5.89
229
5.73
189
4.91
63
6.05
1.7
5.9
6.39
6.11
120.7
6.1
6.6
5.87
6.46
6.27
3.45
4.50
2
5.53
W
(mm)
50
4.9
50
4.9
50
4.9
50
4.9
50
4.9
50
4.9
50
4.9
RESULTS AND DISCUSSION
Fatigue lives of different material at different stress levels, probability of failure and confidence levels are obtained using
the proposed model. In the present paper only few results have been presented. The predicted results are also compared
with the available experimental fatigue life, taken from the literature [1-28].
The fatigue life predictions shown in Figs.1-7 are based on minimum tolerable error of 5%. It is seen that the difference
between the experimental and predicted lives are less at higher stress level compared to lower stress level. This
demonstrates that higher scatter band exists at lower stress level compared to higher stress level. It is seen that in all
cases studied in the present investigation the percentage difference is less than 5% for most of the stress levels and
material.
305
Stress Amplitude(MPa)
Experimental
300
95% Confidence
295
99% Confidence
290
285
280
275
270
265
260
255
150000
200000
250000
300000
350000
400000
450000
Cycles to failure
Fig. 1 S-N curve for 0.26% carbon steel at 50% Probability and different confidence level, R=-1
290
Experimental
Stress Amplitude(MPA)
270
95% Confidence
99% Confidence
250
230
210
190
170
150
0
200000
400000
600000
800000
1000000
1200000
1400000
Cycles to Failure
Fig. 2. S-N curve for 304 stainless steel at 50% Probability and different confidence level, R=-1
250
Experim ental
95% Confidence
Stress Amplitude(MPa)
230
99% Confidence
210
190
170
150
0
1000000
2000000
3000000
4000000
Cycles to failure
Fig. 3. S-N curve for 2024 T3 Aluminum alloy at 50% Probability and different confidence level, R=-1
250
Experimental
50% Probability
Stress Amplitude(MPa)
230
90% Probability
210
190
170
150
0
1000000
2000000
3000000
4000000
Cycles to failure
Fig. 4. S-N curve for 2024 T3 Aluminum alloy at 90% confidence level and different probability, R=-1
1400
Stress Amplitude(MPa)
1200
Experimental
50% Probability
90% Probability
1000
800
600
400
200
0
100000
200000
300000
400000
500000
600000
700000
800000
Cycles to Failure
Fig. 5 S-N curve for 4340 steel at 90% confidence level and different probability, R=-1
350
Experimental
50% Probability
90% Probability
Stress Amplitude(MPa)
300
250
200
150
100
0
50000
100000
150000
200000
250000
300000
350000
400000
Cycles to failure
Fig. 6 S-N curve for 7075-T6 Aluminum alloy at 90% confidence level and different probability, R=-1
650
Experimental
95% Confidence
99% Confidence
Stress Amplitude(MPa)
600
550
500
450
0
300000
600000
900000
1200000
1500000
1800000
Cycles to failure
Fig. 7 S-N curve for Ti-6Al-4V Titanium alloy at 50% Probability and different confidence level, R=-1
The model was used to study the crack growth. The variation of crack growth with number of cycles is shown in figure 811. The model was also verified with the ASTM procedure described in its standard E 739-91. It is found that present
model can be used successfully to find out the S-N curve at different probability and confidence level, only from the
monotonic and fracture properties of the material.
0.044
Crack Length a(m)
0.034
0.024
0.014
Experimental
Predicted
0.004
0
10000
20000
30000
40000
50000
60000
70000
80000
Number of Cycles
Fig. 8 Crack length vs. Number of Cycles Plot at constant maximum load (Pmax=8829N), 90%probability, 90% confidence
level, R=0 data (Kumar & Pandey, 1990)
Crack Length a(m)
0.044
0.034
0.024
0.014
Experimental
Predicted
0.004
0
20000
40000
60000
80000
100000
120000
140000
160000
Number of Cycles
Fig. 9 Crack length vs. Number of Cycles Plot at constant maximum load (Pmax=8829N), 90%probability, 90% confidence
level, R=0.3 data (Kumar & Pandey, 1990)
0.044
Crack Length a(m)
0.034
0.024
0.014
Experimental
Predicted
0.004
0
10000
20000
30000
40000
50000
60000
70000
Number of Cycles
Fig. 10. Crack length vs. Number of Cycles Plot at constant load range (∆P=8829N), 90% probability, 90% confidence
level, R=0 data (Kumar & Pandey, 1990)
0.044
Crack Length a(m)
0.034
0.024
0.014
Experimental
Predicted
0.004
0
10000
20000
30000
40000
50000
60000
Number of Cycles
Fig. 11. Crack length vs. Number of Cycles Plot at constant load range (∆P=8829N), 90% probability, 90% confidence
level, R=0.1 data (Kumar & Pandey, 1990)
Conclusions
The following conclusions can be drawn from the present study.
1. Fatigue life is greatly influenced by the uncertainty associated with material properties and specimen geometry
parameter.
2. Material properties and geometry parameters follows normal distribution.
3. Proposed model can be used to predict fatigue life as well as crack growth rate under given confidence and
probability of failure with tolerable error.
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