1
ME 571 Reliability Based Design
Optional Project , Given 2/13/20, Due 3/12/20
Introduction
Light aluminum alloys are used extensively for structural elements in aerospace and automotive
applications where weight is an important design metric. One such alloy that is commonly used
is the wrought Al 2024-T351 alloy, known for its fatigue strength. A commonly used approach
to model the life of this alloy under low cycle fatigue (less than ~10,000 cycles) is to use the
Coffin – Manson relation (Xue, 2008):
𝛥𝜖𝑝
2
= 𝜖𝑓(2𝑁𝑓 )
𝑐
(1)
where, 𝛥𝜖𝑝 is the plastic strain range to which the material is subjected in each loading cycle
(see Figure 1)
𝑁𝑓 is the number of load cycles to failure (2𝑁𝑓being the total number of reversals from
tension to compression)
and 𝜖𝑓 is the strain to failure under monotonic loading (corresponding to 𝑁𝑓 = 0.5).
Figure 1 Schematic of stress-strain loop for a specimen under cyclic loading. The area inside the
loop is the plastic work dissipated by the material per unit volume per cycle.
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A set of twenty fatigue tests were performed on tension specimens of Al 2024-T351 and the
results are shown in Figure 2. Specimens were subjected to cyclic axial loads until failure or until
the stiffness dropped below a critical value. The loads were large enough to yield the specimens
and deform them plastically, to cause low cycle fatigue. The Coffin-Manson relation was used to
fit the data and the fit is shown as a dashed line. It is observed that the data has an inherent
curvature which the straight-line Coffin-Manson relation is not able to capture, resulting in a
poor fit.
Studies have shown that Eq. (1) is not very accurate for low failure cycles of below 200 cycles
(Xue, 2008). The relation is empirical and not based on foundational physics. Thus, the objective
of this project is to explore different models and develop an alternative approach for
characterizing low cycle fatigue. The models that will be studied here are:
1. Power law (Coffin-Manson) relation
2. Maximum Entropy distributions (Truncated Exponential and Left Truncated Normal)
3. Weibull distribution (also, a Max Entropy distribution, it turns out)
Figure 2: Coffin-Manson relation used to model experimental fatigue data for Al 2024-T351. The
curvature of the data is not captured by the relation
Damage Parameter
The inelastic dissipation in each loading cycle gives a measure of the damage to the specimen.
The material damage is described by a parameter, D, which ranges from 0 (undamaged) to 1
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(failure). Following the famous Miner’s rule, for a test specimen that fails in 𝑁𝑓 cycles, we can
write the damage accumulated over 𝑁 cycles as:
𝐷 =
𝑁
𝑁𝑓
(2)
The damage per each load reversal is then given as:
𝐷𝑟𝑒𝑣 =
𝐷
2𝑁
=
1
2𝑁𝑓
(3)
In this project, we will characterize fatigue failure using the damage parameter and the inelastic
dissipation per load reversal. Figure 1 shows a sample stress-strain loop, obtained from a single
loading cycle of a fatigue test. The area of each loop, 𝐴𝑓 , gives the plastic dissipation in each
cycle (or 2 load reversals).
𝑊𝑓
𝑁𝑓
= 𝐴𝑓 (4)
where, 𝑊𝑓 is the plastic dissipation in the material
and 𝑁𝑓 is the number of cycles to failure (2𝑁𝑓 is the number of load reversals)
Power Law Relation
One of the simplest models that can be applied is a power law relation between the damage
and the plastic dissipation per load cycle. The Coffin-Manson relation can be rewritten in terms
of the damage parameter to obtain one such relation. Rewriting Eq. (1), we get:
𝐷𝑟𝑒𝑣 =
1
2𝑁𝑓
= (
𝛥𝜖𝑝
2𝜖𝑓
)
−
1
𝑐
(5)
This suggests an analogous power-law model in terms of the inelastic dissipation:
𝐷𝑟𝑒𝑣 = 𝑘 (
𝑊𝑓
2𝑁𝑓
)
−
1
𝑐
(6)
where, 𝑊𝑓 is the inelastic dissipation in the material (area inside the loop in Figure 1),
𝑁𝑓 is the number of cycles to failure (2𝑁𝑓 is the number of load reversals)
and 𝑘 is a constant related to the critical inelastic dissipation for monotonic failure.
Eq. (6) can now be used to model low cycle fatigue.
Weibull Distribution
The Weibull distribution has the form:
𝑓(𝑡; 𝜂, 𝛽) = {
𝛽
𝜂
(
𝑡
𝜂
)
𝛽−1
𝑒
−(
𝑡
𝜂
)
𝛽
, 𝑡 ≥ 0
0 , 𝑡 < 0 (7) where 𝜂 and 𝛽 are positive and called the characteristic life (scale) and shape parameters, respectively. The corresponding CDF has the form: 4 𝐹(𝑡; 𝜂, 𝛽) = {1 − 𝑒 −( 𝑡 𝜂 ) 𝛽 , 𝑡 ≥ 0 0 , 𝑡 < 0 (8) Maximum Entropy Distributions Shannon (Shannon, 1948) proposed an information function, 𝐼(𝑝), that quantifies the data content of an event. For an event with probability 𝑝: 𝐼(𝑝) = − ln 𝑝 (9) For a set of 𝑛 probable events, the expected value of the information function has a form analogous to the Gibb’s entropy function (Jaynes, 1957) used in thermodynamics, and is hence called Shannon’s information entropy function, 𝐻(𝒑): 𝐻(𝒑) = 𝐸(𝐼(𝑝)) = − ∑ 𝑝𝑖 ln 𝑝𝑖𝑖 (10) Jaynes (Jaynes, 1957) proposed that the best choice of a distribution, in the absence of any other information, is one that maximizes Eq. (10) subject to the constraint: ∑ 𝑝𝑖𝑖 = 1 (11) Such a distribution will have the least amount of bias and will hence provide a superior fit. The distribution can be computed by solving an optimization problem, with additional information about the distribution (e.g., the mean) added as constraints. The maximum entropy principle can be applied to the Weibull distribution discussed in the previous section to obtain unbiased models for fatigue modeling. The entropy function of the Weibull distribution can be evaluated from Eqs. (7) and (10), and is given by: 𝐻(𝑓) = 𝛾 (1 − 1 𝛽 ) + ln 𝜂 𝛽 + 1 (12) where 𝛾 ≈ 0.577 is called the Euler-Mascheroni constant. Truncated Exponential Distribution Maximizing Eq. (12) with the constraint of the mean fixed at 𝐸(𝑓(𝑡)) = 𝜇 gives the exponential distribution: 𝑓(𝑡) = { 1 𝜇 𝑒 − 𝑡 𝜇 , 𝑡 ≥ 0 0 , 𝑡 < 0 (13) Since the damage parameter which is being modeled reaches unity at a finite number of cycles (or the reliability will be zero sooner than 𝑡 = ∞), a truncated form of Eq. (13) would be more appropriate. If the truncation occurs at 𝑡 = 𝑎 > 0, the truncated distribution is given by:
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𝑓𝑡𝑟(𝑡) = {
1
𝐶
1
𝜇
𝑒
−
𝑡
𝜇 , 0 ≤ 𝑡 ≤ 𝑎
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝐹𝑡𝑟(𝑡) = {
0 , 𝑡 < 0 1 𝐶 (1 − 𝑒 − 𝑡 𝜇 ), 0 ≤ 𝑡 ≤ 𝑎 1, 𝑡 > 𝑎
(14)
where, 𝐶 = (1 − 𝑒
−
𝑎
𝜇 ) is a correction factor for the truncation. Note that 𝜇 is the mean of the
parent exponential distribution and not for the truncated distribution.
Left Truncated Normal Distribution
Study shows that with proper choice of moment functions, a truncated normal distribution can
also be derived from the maximum entropy principle. Young et al (Young & Subbarayan, 2019)
provided the following probability density function for left truncated normal distribution which
is truncated at t = 0 shown in Figure 3:
𝑓𝑡𝑟𝑢𝑛𝑐 (𝑡; 𝜇, 𝜎) = {
(
1
1−𝐹𝑛𝑜𝑟𝑚(0,𝜇,𝜎)
)
1
𝜎√2𝜋
𝑒
−
𝑡−𝜇2
2𝜎2 , 𝑡 > 0
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(15)
and cumulative distribution function:
𝐹𝑡𝑟𝑢𝑛𝑐 (𝑡; 𝜇, 𝜎) = {
𝐹𝑛𝑜𝑟𝑚(𝑡,𝜇,𝜎)−𝐹𝑛𝑜𝑟𝑚(0,𝜇,𝜎)
1−𝐹𝑛𝑜𝑟𝑚(0,𝜇,𝜎)
, 𝑡 > 0
0 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
(16)
where the factor in the denominator of the CDF is the area correction factor to make up the
density lost for 𝑡 < 0.
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Figure 3: Left Truncated normal distribution plotted with the parent (non-truncated) normal
distribution
Problem Statement
Twenty low cycle fatigue tests were performed on aluminum tension specimens of Al 2024-
T351, and the life to failure is summarized in Table 1. Stress-strain curves similar to Figure 1
were obtained, and the inelastic dissipation per load reversal was estimated from the area
enclosed in the stress loops and is listed in Table 1.
Question 1: Four different fatigue models have been proposed to describe the LCF failure data
listed in Table 1. These are:
i. Coffin-Manson based power law relation (Eq. (6))
ii. Weibull distribution (Eq. (8))
iii. Truncated Exponential distribution (Eq. (14))
iv. Left Truncated Normal distribution (Eq. (16))
Fit each of the above models (CDF for ii, iii and iv) to the given fatigue failure data. Summarize
your model fit parameters in a table. Discuss how well each model fits the data. Compute the
sum of the squared error (sum of the square of the difference between the fit and the data
point at each 𝑁𝑓 in Figure 2). Which model would you choose to best capture the features of
the data? Justify your response.
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Question 2: Since there are only 20 data points, the Kolmogorov-Smirnov test is the preferred
test for goodness-of-fit. Use the K-S test to estimate the goodness of fit and level of significance
for the fits of Q1.
Question 3: A engine cylinder head made of aluminum (Al 2024-T351) is to be designed for
fatigue resistance. A tension specimen made of the material is subjected to 10 cycles of load
similar to that experienced in the real engine and the inelastic dissipation computed from the
stress-strain loops is 102.3 MPa per load reversal. Estimate the fatigue life of the engine
cylinder head using the models and estimated parameters of Q1.
Question 4: A set of 20 components were tested under operational use conditions. The test
was stopped once 10 components failed. The number of cycles to failure for the failed
components are listed in Table 2. Post-failure analysis using a scanning electron microscope
(SEM) found that some of the failed components contained relatively large impurities from
which fatigue cracks initiated. The components with impurities are marked in Table 2 with a √
symbol. Determine the mean time to failure for samples that contained impurities.
Report
Document the fitting results and errors in a neatly written text or typed report. Include a single
plot containing the final fits for each of the fatigue models. Give a brief summary of the
concepts learned in this project. The written report should be no more than two pages long
with additional tables and figures included as an appendix. The report should be single spaced
and use 12 pt font.
Grading rubric: Q1 50 pts, Q2 10 pts, Q3 10 pts, Q4 20 pts, Report 10 pts.
Hint: Fitting the Truncated Exponential Function
The fit for the truncated exponential function is nonlinear, and the equation is valid only for 0 <
x < a. The following tips may help with the fitting process:
1. In order to reduce the degree of nonlinearity, define
1
𝜇
as a new parameter (say 𝑚).
E.g. change −
𝑥
𝜇
into 𝑚𝑥. This improves the stability of the fitting algorithm leading
to better fits.
2. One way to handle the truncation is to define a function in MATLAB that takes in
𝑥, 𝑎, 𝑚 as input and returns 1 if 𝑥 > 𝑎 or evaluates Eq. (14) for 0 < 𝑥 < 𝑎 (such as
myFunc(x,a,m)). This function can then be used to define the fit type as:
ftype = fittype('myFunc(x,a,m)');
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3. To improve the stability of the fitting algorithm, the robust fitting option can be
enabled as:
fit(x,y,ftype,'Robust','on')
When robust fitting is enabled, MATLAB assigns smaller weights to the input data
outliers which reduces their influence on the fit. This can also improve the
convergence of the fit.
Table 1: Experimental data from 20 low cycle fatigue tests on Al 2024-T351 tension specimens
Test No
Fatigue Life
(𝟐 𝑵𝒇)
Inelastic Dissipation per
Reversal (𝑾𝒇/𝟐𝑵𝒇)
Damage per Cycle
(𝑫𝒓𝒆𝒗 = 𝟏/𝟐𝑵𝒇)
1 1 292.032 1
2 1 424.008 1
3 1315 45.2088 0.00076046
4 649 33.4152 0.00154083
5 2156 31.4496 0.00046382
6 2506 30.0456 0.00039904
7 3312 26.3952 0.00030193
8 1981 21.06 0.0005048
9 7692 20.0772 0.00013001
10 9795 13.87152 0.00010209
11 16102 10.44576 6.2104E-05
12 9024 8.64864 0.00011082
13 18905 8.17128 5.2896E-05
14 13999 7.94664 7.1434E-05
15 10916 7.94664 9.1609E-05
16 49039 3.42576 2.0392E-05
17 28156 3.53808 3.5516E-05
18 102651 1.367496 9.7417E-06
19 286190 0.2552472 3.4942E-06
20 409951 0.0693576 2.4393E-06
Table 2: Number of cycles to failure for the failed components in Q4.Components with the impurities are
marked with √.
Test No. Failure Life Impurity
1 40 √
2 67 √
3 83 √
4 257 ×
5 359 √
6 470 √
7 712 ×
9
8 824 ×
9 905 √
10 1109 ×
References
1. Jaynes, E. (1957). Information theory and statistical mechanics. Physical Review, 106(4),
620-630.
2. Shannon, C. (1948). Mathematical theory of communication. Bell System Technical
Journal, 27(3-4), 379-423, 623-656.
3. Xue, L. (2008). A unified expression for low cycle fatigue and extremely low cycle fatigue
and its implication for monotonic loading. International Journal of Fatigue, 30, 1691-
1698.
4. Young, C., & Subbarayan, G. (2019). Maximum entropy models for fatigue damage in
metals with application to low-cycle fatigue of Aluminum 2024-T351. Entropy, 21, 967.