Vibration fatigue is describing the engineering material fatigue which is caused by forced vibration of random nature. Typically, the excited structure is usually subjected to arbitrary variable amplitude loading spectrums in the fatigue weakness area. Due to high nonlinearity of the fatigue damage evolution, it is extremely difficult to analyze the fatigue process under the vibration loading by cycle-based method in the time-domain. So, in this paper a time-based method is proposed to analyze the fatigue crack growth behavior under arbitrary variable amplitude loading. This approach is based on the crack tip opening displacement (CTOD) estimation, crack closure model and the fatigue damage evolution function. A set of fatigue experimental data for Al7075-T6 under “Christmas Tree” loading, a typical vibration loading spectrum, is used to validate the method. The predictions of fatigue life under arbitrary load spectra match the testing data very well. Moreover, the simulation under a random loading is employed to demonstrate the verification of this formulation

Shan Jiang

Wei Zhang

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 229 
A time-based model for random vibration fatigue 
analysis 
Shan Jiang1, Wei Zhang2 
Science and Technology on Reliability and Environmental Engineering Laboratory,  
School of Reliability and Systems Engineering, Beihang University, Beijing, China 
2Corresponding author 
E-mail: 1jshan.susan@gmail.com, 2zhangwei.dse@buaa.edu.cn 
(Accepted 15 September 2014) 
Abstract. Vibration fatigue is describing the engineering material fatigue which is caused by 
forced vibration of random nature. Typically, the excited structure is usually subjected to arbitrary 
variable amplitude loading spectrums in the fatigue weakness area. Due to high nonlinearity of 
the fatigue damage evolution, it is extremely difficult to analyze the fatigue process under the 
vibration loading by cycle-based method in the time-domain. So, in this paper a time-based 
method is proposed to analyze the fatigue crack growth behavior under arbitrary variable 
amplitude loading. This approach is based on the crack tip opening displacement (CTOD) 
estimation, crack closure model and the fatigue damage evolution function. A set of fatigue 
experimental data for Al7075-T6 under “Christmas Tree” loading, a typical vibration loading 
spectrum, is used to validate the method. The predictions of fatigue life under arbitrary load 
spectra match the testing data very well. Moreover, the simulation under a random loading is 
employed to demonstrate the verification of this formulation. 
Keywords: crack propagation, vibration fatigue, CTOD, variable amplitude loading. 
1. Introduction 
Vibration fatigue theory has been introduced to deal with the fatigue behavior under any kinds 
of arbitrary vibration loading. Many structures components are frequently subjected to vibration 
loading of random nature. Since fatigue is one of the predominant mechanical failure causes, 
fatigue life prediction has become a necessary subject in many researches [1-3]. The approach of 
predicting fatigue life could be divided into frequency domain and time domain. The time domain 
methods are based on the cycle counting and cumulative damage rules [4]. The identification of 
cycles is usually carried out through the rainflow counting method [5]. And then the Miner damage 
accumulation rule is used to perform the accumulation of damage. The time domain has been 
widely accepted to predict the fatigue life. However, when a component is subject to variable 
amplitude loading, the cycle calculation will be very time consuming. In addition, the variable 
load interaction has the influence on the fatigue crack growth behavior which the most of the 
existing time domain methods cannot account for [6]. Therefore, our investigation in this paper 
focuses on the time-based model development aim to predict the fatigue crack behavior of 
components subjected to arbitrary variable amplitude loading.  
Many models have been proposed to predict the fatigue crack propagation under variable 
amplitude loading. Elber proposed the approach to describe the crack growth behavior, which 
considers crack closure [7]. Wheeler [8] and Willenborg [9] presented a series of models based on 
the plastic zone size ahead of the crack tip. The fatigue crack growth computer program developed 
by Forman et al. could be used to estimate for constant and variable amplitude loading [10]. Most 
studies involve the nonlinear analysis of cyclic plasticity and crack contact analysis, which causes 
the computing process of the random variable amplitude loadings situation extremely  
complicated. 
In many recent studies, the CTOD variation concept has been indicated to correlate with the 
fatigue crack growth rate under cyclic loadings [11, 12]. Liu, Lu and Xu proposed a theoretical 
CTOD formulation under variable amplitude loading [14]. In addition, Zhang and Liu performed 
a state-of-art in-situ SEM testing and presented a virtual crack annealing model to calculate the 
A TIME-BASED MODEL FOR RANDOM VIBRATION FATIGUE ANALYSIS.  
SHAN JIANG, WEI ZHANG 
230 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
crack closure which has the effect on the CTOD variation [12, 13]. Based on that, Zhang and Liu 
developed a time-based fatigue crack growth model to predict the fatigue crack behavior under 
variable amplitude loading, which combine the CTOD variation estimation with the crack closure 
model [15]. Through this model, the CTOD variation at any loading level can be obtained. 
The paper is organized as follows. First, the model to calculate the CTOD variation which is 
combined with the crack closure model and the fatigue damage evolution function is briefly 
derived. Based on this method, the fatigue crack growth rate can be estimated. Next, the fatigue 
testing of Al 7075-T6 under “Christmas Tree” loading has been performed to verify this 
formulation. By using the “Christmas Tree” loading, the high ratio and low ratio situation will be 
discussed. The predictions of the proposed approach match the experimental data very well. 
Moreover, the method will be simulated under random loading. Finally, some conclusions are 
given based on the current investigation. 
2. Methodology 
It is proven that CTOD variation correlates with the fatigue crack propagation. The analytical 
solution of CTOD could be expressed as [16]: 
ߜ = ܭ
ଶ
ܧߪ௬
= ߪ
ଶߨܻܽଶ
ܧߪ௬
, (1)
where ܧ is the Young’s modulus, and ߪ௬ is the yield strength. ܭ is the stress intensity factor (SIF), 
which can be expressed as [17]: 
ܭ = ߪ√ߨܽ ⋅ ܻ,
ܻ(ܽ/ݓ) = 1.12 − 0.23(ܽ/ݓ) + 10.55(ܽ/ݓ)ଶ − 21.22(ܽ/ݓ)ଷ + 30.39(ܽ/ݓ)ସ, (2)
where ܽ is the crack length and ݓ is the plate width. 
A general mathematic calculation of CTOD under random cyclic loading is derived in paper 
[14]. And then the crack closure is considered. It is proven that the crack closure is an important 
factor of the fatigue crack growth behavior through the fatigue testing and theoretical analysis 
[12, 13]. Therefore, Zhang and Liu modified the formulation above [15]. Based on a stationary 
crack, the mathematic estimation of CTOD variation with crack closure during the reloading path 
and unloading path can be expressed as [15]: 
ߜ௟௢௔ௗ௜௡௚ 
     =
ە
ۖۖ
۔
ۖۖ
ۓ ܭ
ଶ
ܧσ௬
, ܭ > ܭ୫ୟ୶,௠௘௠,
ߜ௠௜௡,௠ିଵ +
(ܭ − ݉ܽݔ(ܭ௠௜௡,௠ିଵ, ܭ௥௘௙,௠ିଵ))ଶ
2ܧߪ௬
ܪ൫ܭ − ܭ௥௘௙,௠ିଵ൯,
ܭ௠௔௫,௠௘௠ ≥ ܭ,
ܭ ≥ ܭ௠௔௫,௠ିଵ,
ߜ௠௜௡,௠ +
(ܭ − ݉ܽݔ(ܭ௠௜௡,௠, ܭ௥௘௙,௠))ଶ
2ܧߪ௬
ܪ൫ܭ − ܭ௥௘௙,௠൯, ܭ୫ୟ୶,௠ିଵ ≥ ܭ,
 
ߜ௨௡௟௢௔ௗ௜௡௚ 
     =
ۖە
۔
ۖۓߜ௠௔௫,௠ିଵ −
(ܭ௠௔௫,௠ିଵ − ݉ܽݔ (ܭ, ܭ௥௘௙,௠ିଵ))ଶ
2ܧߪ௬
ܪ൫ܭ௠௔௫,௠ିଵ − ܭ௥௘௙,௠ିଵ൯,   ܭ ≤ ܭ௠௜௡,௠ିଵ,
ߜ௠௔௫,௠ −
(ܭ௠௔௫,௠ − max(ܭ, ܭ௥௘௙,௠ିଵ))ଶ
2ܧߪ௬
ܪ൫ܭ௠௔௫,௠ − ܭ௥௘௙,௠൯, ܭ ≥ ܭ௠௜௡,௠ିଵ,
 
(3)
where ܭ௥௘௙,௠  indicates the crack closure loading level at the ݉ th peak or valley. ܪ  is the 
Heaviside step function. If the ܭ or ܭ௠௔௫ is smaller than ܭ௥௘௙, the crack will close and CTOD 
A TIME-BASED MODEL FOR RANDOM VIBRATION FATIGUE ANALYSIS.  
SHAN JIANG, WEI ZHANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 231 
will remain constant.  
All the derivation above is based on the stationary crack. For the practical crack, the total 
CTOD variation considering the crack growth can be written as [15]: 
݀δ = ∂δ∂σ ݀σ +
∂δ
∂ܽ ݀ܽ.
(4) 
Therefore, the general calculation for CTOD can be expressed as [15]: 
݀ߜ௟௢௔ௗ௜௡௚ 
     =
ە
ۖۖ
۔
ۖۖ
ۓ2ߪߨܻܽ
ଶ
ܧߪ௬
݀ߪ + ߪ
ଶߨܻଶ
ܧߪ௬
݀ܽ, ߪ > ߪ௠௔௫,௠௘௠,
൫ߪ − ݉ܽݔ൫ߪ௠௜௡,௠ିଵ, ߪ௥௘௙,௠ିଵ൯൯ߨܻܽଶ
ܧߪ௬
݀ߪ +
(ߪ − ݉ܽݔ (ߪ௠௜௡,௠ିଵ, ߪ௥௘௙,௠ିଵ))ଶߨܻଶ
2ܧߪ௬
݀ܽ,
ߪ௠௔௫,௠ିଵ ≥ ߪ,
0,                                                             ܪ൫ߪ − ߪ௥௘௙൯ = 0,
 
݀ߜ௨௡௟௢௔ௗ௜௡௚ =
ە
ۖ
۔
ۖ
ۓ−
൫ߪ௠௔௫,௠ିଵ − ݉ܽݔ൫ߪ, ߪ௥௘௙,௠ିଵ൯൯ߨܻܽଶ
ܧߪ௬
݀ߪ,    ߪ ≤ ߪ௠௜௡,௠ିଵ,
−
൫ߪ௠௔௫,௠ − ݉ܽݔ൫ߪ, ߪ௥௘௙,௠ିଵ൯൯ߨܻܽଶ
ܧߪ௬
݀ߪ,        ߪ ≥ ߪ௠௜௡,௠ିଵ,
0,       ܪ൫ߪ௠௔௫ − ߪ௥௘௙൯ = 0.
 
(5)
In order to predict the fatigue crack propagation, the simple mathematical model is present in 
paper [15]. The instantaneous crack growth rate can be expressed as [15]: 
݀ܽ
݀ݐ = ݂(ܭ୫ୟ୶, Δܭ, ߜ) = ቐ
ܣ ⋅ Δܭ ⋅ ܭ௠௔௫
2ඥߜ − ߜ௥௘௙
݀ߜ
݀ݐ , ߜ − ߜ௥௘௙ > 0,
0, otherwise,
(6) 
where ܣ is a fitting parameter. Based on the formulation above, the crack length incremental at 
any point during the cyclic loading can be written as [15]: 
Δܽ = න
ఙ೟భ
ఙ೟బ
݂(ܭ௠௔௫, ߂ܭ, ߜ)݀ݐ. (7) 
3. Model validation 
3.1. Model validation using “Christmas Tree” loading spectrum 
In this section, the model above will be validated through comparing the predictions with the 
experimental data. The Fig. 1 shows the type of vibration loading spectrum, which could be 
considered as the superposition of two loading spectra.  
Therefore, the “Christmas Tree” loading spectrum is employed to verify the model. Zhang and 
Liu provided fatigue test data for edge-cracked 7075-T6 aluminum alloy specimen under variable 
amplitude loading [15]. The geometric dimensions of the specimen are respectively:  
thickness = 4.7 mm, width = 40 mm. The geometric factor of the stress intensity factor for this 
specimen is: 
A TIME-BASED MODEL FOR RANDOM VIBRATION FATIGUE ANALYSIS.  
SHAN JIANG, WEI ZHANG 
232 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
ܻ = 1
√ߨߙ
ቆ 2 + ߙ(1 − ߙ)ଷ ଶ⁄ ቇ (0.886 + 4.64ߙ − 13.32ߙ
ଶ + 14.72ߙଷ − 5.6ߙସ). (8)
 
Fig. 1. Schematic illustration of the vibration loading spectrum 
 
a) Baseline loading spectrum 
 
b) 20 % high ratio spectrum 
 
c) 10 % low ratio spectrum 
Fig. 2. “Christmas tree” loading profile 
Fig. 3. Comparison of the model predictions with 
experimental results 
Fig. 4. Schematic of the variable amplitude loading 
 
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 104
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
Time
Cr
ac
k l
en
gth
 a(
m)
 
Baseline Test data
20% High Ratio Test data
10% Low Ratio Test
Baseline Predtion
20% High Ratio Predtion
10% Low Ratio Predtion
0 2000 4000 6000 8000 10000 12000 14000 160000
5
10
15
20
25
30
35
40
K 
MP
a*m
0.5
Time Point
A TIME-BASED MODEL FOR RANDOM VIBRATION FATIGUE ANALYSIS.  
SHAN JIANG, WEI ZHANG 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 233 
The experimental loading spectra are shown in Fig. 2. The typical arbitrary load spectrum 
situation is included in the “Christmas Tree” spectra. The vibration is treated as loading spectrum 
to affect the fatigue crack propagation behavior. 
The model predictions are compared to the test data in Fig. 3. It is obvious that the predictions 
match the experimental data very well. The present formulation provides a good estimate of 
fatigue crack growth under arbitrary amplitude loading. 
3.2. Model predictions under random loading 
In this section, the simulation of the model above will be illustrated. The variable amplitude 
loading spectrum is shown in Fig. 4.  
According to the Eq. (5), the CTOD variation under variable amplitude loading is calculated 
and shown in the Fig. 5. When the ܭ is less than ܭ௥௘௙, the crack is closed and the CTOD remains 
constant. 
More detailed information can be obtained in the ܰ-ܽ curve in Fig. 6, which shows the crack 
growth rate under the variable amplitude loading. 
Fig. 5. CTOD variation under variable amplitude 
loading 
Fig. 6. Simulation of the model 
 
4. Conclusions 
The fatigue crack growth behavior under random vibration loading is usually accounted for by 
analyzing the fatigue properties. The vibration is treated as the loading history or loading spectrum 
to affect the crack propagation. Therefore, the time-based fatigue crack growth model is described 
in this paper to predict the fatigue crack growth under variable amplitude loading. This model is 
based on the CTOD variation, which is combined with the crack closure. The following 
conclusions can be drawn: 
The time-based method is proven to predict the fatigue life under variable amplitude loading 
easily, accurately and successfully through the fatigue testing and the simulation validation. 
Furthermore, this model avoids complex calculation process and cycle counting, which is different 
from the existing method. 
The model validation using “Christmas Tree” is performed and very good correlations are 
observed between model predictions and experimental observations. 
During the vibration cycles, the loading frequency has effects on the crack growth. However, 
the frequency factor is not considered in this investigation. In the future, the method should be 
extended by taking account of the frequency influence. 
0 5 10 15 20 25 30 35 402
4
6
8
10
12
14
16
18
20
22
K MPa*m0.5
CT
OD
 um
0 2000 4000 6000 8000 10000 12000 14000 160005.0005
5.001
5.0015
5.002
5.0025
5.003
5.0035
5.004 x 10
-3
Time Point
cra
ck
 le
ng
th 
m
A TIME-BASED MODEL FOR RANDOM VIBRATION FATIGUE ANALYSIS.  
SHAN JIANG, WEI ZHANG 
234 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
Acknowledgements 
The research is financially supported by the National Natural Science Foundation of China 
No. 51405009. 
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A time-based model for random vibration fatigue analysis

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A time-based model for random vibration fatigue analysis

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