Artificial Neural Network Model for Prediction of Fatigue Lives of
Composites Materials
Sanjay Mathur** Prakash Chandra Gope* and J. K. Sharma*
*Department of Mechanical Engineering,
** Department of Electronics & Communication Engineering
College of Technology, G B Pant University of Agriculture & Technology,
Pantnagar-263145, Udham Singh Nagar, Uttaranchal, India
Email: [email protected], Mobile 91 9411159916, Fax: 91 5944 233338
ABSTRACT
In the present study an Artificial Neural Network (ANN) model is developed for fatigue life prediction of
carbon fibre reinforced plastics (CRPF) IM7/977, HTA/913, T800/5245, T800/924 with [(±45,02)2]S lay up and
HTA/919 with [(±45,90,0)2]S lay up and glass fibre composites G/913, G913/913, G913/920, G913/SiC,
G913/PE, G913/G laminates. The ANN data base contains more than five hundred fatigue lives over a range of
different stress ratios of 10 to -3.33. Different monotonic, fatigue and statistical properties have been taken as
input parameter. The inputs to ANN model are lay up, volume fraction, monotonic properties such as tensile
modulus, tensile strength, compression strength, failure strain, applied load parameters such as stress ratio,
maximum stress, minimum stress, probability of failure and statistical parameters of fatigue life. The output of the
ANN is logarithmic value of fatigue life cycles. The architecture selected has two hidden layers with 18 and 6
units each. The learning rate coefficient is optimised as 0.9 and momentum factor as 0.3. The error between
experimental and predicted is found to be less than 5% in most of the cases.
INTRODUCTION
The application of composites as engineering materials has become state of art and fatigue is one of the most
complicated problems for fibre composites. The fatigue life prediction of a newly developed material is costly and
time consuming. A potential solution to this problem is offered by artificial neural networks (ANNs) [1-15].
In recent years ANN have been used for the prediction of fatigue lives of composite materials. Assaf et al [2]
applied the ANN approach to predict the fatigue life of unidirectional glass fiber composite materials. Lee et al
[14] carried out an ANN prediction on carbon/glass fiber reinforced plastic laminate. In their study they have used
only one lay up to evaluate a possible ANN structure. Recently, Zhang et al [15] presented a review paper on the
application of neural network to polymer composites. They have suggested that further improvement in this
technique is required to find out the correlations between measured parameters and complex properties. Assaf et
al., [3-4] considered various types of ANN such as modular, self-organizing, radial basis, and principle
component analysis networks for improving the prediction accuracy. A comparison of such ANN structures in
predicting fatigue behaviour of unidirectional glass fiber/epoxy composite laminate for various fiber orientation
angles and stress ratios is investigated. The work showed that, compared to the classical feed forward neural
network, other types of ANN could be used to improve the fatigue life prediction of composite materials.
Modeling of material behavior generally involves the development of a mathematical model derived from
observations and experimental data. Ince., [16] used an alternative way i.e. back propagation artificial neural
network (ANN) to model Two-Parameter Model (TPM) in the fracture of cementations material. The results of an
ANN-based TPM look viable and very promising. Artificial neural networks (ANNs) were used to predict the
residual strength of glass fiber-reinforced plastic beams pre-fatigued in flexure up to different portions of fatigue
life. The acoustic emission signals recorded during the tests for the measurement of residual strength, and the
associated applied stress, were provided as input by Iorio et al., [6]. An optimization of the network configuration
was carried out, using the root-mean square error calculated in the training stage as the optimization parameter.
The predictive accuracy of the optimized ANN, consisting of two nodes in the input layer, four nodes in the hidden
layer, and a single node in the output layer, was tested out by the “leave-out” method. From the results obtained,
ANN provide quiet reliable predictions when the applied load was sufficiently far from the failure load, performing
better than a previous theoretical model, relying on fracture mechanics concept. Therefore, ANN is shown to be
a valid tool in the non-destructive evaluation of composite materials employed in fatigue-sensitive applications.
ANN MODEL
The back propagation algorithm is implemented and a code has been developed in C++ language. The algorithm
is a generalization of least mean square algorithm. The learning rate coefficient (η) and momentum factor (α) are
chosen with number of hidden layers and number of units in different hidden layers to reduce the normalized
error of system to an acceptable level. In the present investigation different network architectures have been
tried. Figs 1-4 shows the variation of normalized system error with different ANN structure parameters. On the
basis of these observations different factors are optimised and used in the modelling of ANN architecture to
estimate fatigue life. Different ANN parameters of the network, converged to good local minima, is presented in
Table 1.
0.0032
Normalized system error
Normalized system error
0.0016
0.0015
0.0014
0.0013
0.0012
0.0011
0.001
0.0009
0.0028
0.0024
0.002
0.0016
0.0012
0.0008
0.1 0.2
13 14 15 16 17 18 19 20 21 22 23 24
0.3 0.4 0.5 0.6
Fig.1 Variation of normalized system error with
number of units in first hidden layer (10,000
iterations, η=0.9, α =0.3, number of units in second
hidden layer = 6).
Fig 3 Variation of normalized system error with
learning rate coefficient, η (α=0.3, number of units
in first hidden layer =18, number of units in second
hidden layer =6 for 10,000 iterations).
0.02
Normalized system error
0.0016
Normalized system error
0.7 0.8 0.9
Learning rate coefficient
Number of units in first hidden layer
0.0015
0.0014
0.0013
0.0012
0.0011
0.001
0.0009
3
4
5
6
7
8
9
0.016
0.012
0.008
0.004
0
0.1
0.2
0.3
Number of units in second hidden layer
Fig 2 Variation of normalized system error with
number of units in second hidden layer (10,000
iterations, η=0.9, α=0.3, number of units in first
hidden layer = 18).
0.4
0.5
0.6
0.7
0.8
0.9
Momentum factor
Fig 4 Variation of normalized system error with
momentum factor, α (η=0.9, number of units in first
hidden layer =18, number of units in second hidden
layer =6 for 10,000 iterations).
Table 1. ANN architecture
Number of hidden
layer
2
Number of units in
hidden layer
18,6
Learning rate
coefficient
0.9
Momentum factor
0.3
Number of iterations
90,000
RESULTS AND DISCUSSION
The fatigue life predictions of carbon fiber reinforced plastics (CFRP) laminates and glass fiber composites have
been carried out by artificial neural networks. Extensive data base of fatigue lives for common CFRPs i.e.
IM7/977, HTA/913, T800/5245, T800/924 with [(±45,02)2]S lay up and HTA/919 with [(±45,90,0)2]S lay up and
glass fiber composites G/913, G913/913, G913/920, G913/SiC, G913/PE, G913/G laminates were used to
evaluate possible ANN architectures. The ANN data base contain more than five hundred fatigue lives over a
range of different stress ratios (R) of 10 to -3.33. Different monotonic, fatigue and statistical properties have been
taken as input parameter. The optimum architecture of ANN (two hidden layer with 18 and 6 nodes in first and
second layer, learning rate parameter η =0.9, momentum factor α = 0.3) have been used for training with 80
percent of data. At an interval of each ten thousand iterations of training the structure was tested with remaining
20 percent of the data set. The predicted and experimental fatigue life results for training is shown in Fig.5 . The
maximum normalized system error (N.S.E.) of system was found to be 0.000467 at 90,000 iterations.
Fig. 6 shows the experimental and predicted fatigue lives for composite T800/5245 at 5 percent and 50 percent
probability of failure. It is seen that accuracy in case of 5 percent is more than 50 percent probability of failure.
This shows that artificial neural networks can be used successfully to predict at a very low level of probability
which is desirable for a designer in critical design area such as aircraft. However, in both the cases prediction is
reasonably accurate. Fig. 7 shows the experimental and predicted fatigue lives for HTA/919 composite materials
for different stress ratio of 10, -3.33, -0.3 and 0.1. Fig .7 shows that at the stress ratio 10 and -3.33 the predicted
and experimental values are very close to each other than other cases. The percentage error between
experimental and predicted fatigue life is found to be less than 5 percent where sufficient data have been
provided during training for a given stress ratio. It has also been observed that due to insufficient number of data,
the error increases. As for this material only one data is been used at stress level 0.65 GPa and stress ratio R = 0.3 the error for this case is found to be 7 percent. Fig.8 shows the variation of experimental and predicted
fatigue life with peak stress for different values of R for composite material IM7/977. Fig.8 shows that due to
higher scatter in input data the predicted life also shows scattered results. In this case the error was found to be
between 10 to 29 percent between experimental and predicted values for the case where standard deviation of
input life data is more or only one data per stress level have been used in the training of the ANN structure. Out
of 15 stress level only in one case the error is 29 percent, in two case the error is about 10 percent and for others
the error remained below 4 percent. Fig. 9 shows the experimental and predicted fatigue lives at different stress
ratio for composite material HTA/913. The results are quiet close to each other at all stress ratios R= 10, -1.5,-1,0.3 and 0.1 respectively. Out of 19 stress levels used the error is about 10 percent for 3 stress levels and for the
remaining stress level, the error remains below 4 percent. In most of the cases ( 11 stress levels) the error
remains below 2 percent. Fig. 10 shows the experimental and predicted fatigue lives for composite material
T800/924. The error remains below 5 percent for most of the stress levels except very few. Higher error has been
found for those cases where very few life data have been used. The difference between experimental and
predicted value at stress level R = 10 is more compared to other stress level. At other stress levels predicted
results are very close to experimental values.
To study the effect of variability of fatigue life on prediction by ANN model Fig. 11 shows the effect of standard
deviation of input fatigue life on the predicted fatigue life. Increasing trends of error between experimental and
predicted values with the standard deviation have been seen. It is observed that due to higher coefficient of
variation in input fatigue life for a given stress level and stress ratio, the scatter in predicted results are more and
hence error between experimental and predicted results are more. This statistical aspect of the fatigue life data
may be used in training and optimization of ANN architecture to get better predicted values.
To study the effect of monotonic properties of material on predicted lives by ANN, Fig. 12 shows the variation of
error with strength factor (the strength factor is defined as ratio of compressive strength to tensile strength). It is
observed that as strength factor increases the percentage error also increases. A linear relationship between
strength factor and percentage error have been observed.
Fig. 13 shows the effect of volume fraction (vf) on percentage error obtained from predicted and experimental
fatigue lives for all investigated materials. An inverse linear relationship between percentage error and volume
fraction have been observed. As the volume fraction of the laminates increases the error decreases.
Log fatigue life, cycles
(ANN)
Training
7
6
5
4
3
2
2
3
4
5
6
7
Log fatigue life, cycles (Experimental)
Fig.5 Variation of experimental and predicted fatigue life by ANN (with 90,000 iterations)
Peak stress, GPa
0.8
0.7
p=50% (Expt.)
P=50% (ANN)
p=5%(Expt.)
p=5% (ANN)
0.6
0.5
0.4
2
3
4
5
6
Log fatigue life, cycles
Fig. 6 Variation of fatigue life with peak stress at R = -1 for T800/5245 for different probabilities of failure.
0.8
0.6
R=10( Expt.)
R=10 (ANN)
Peak stress, GPa
0.4
R=-3.33 (Expt.)
R=-3.33 (ANN)
R=-0.3(Expt.)
0.2
0
R=-0.3 (ANN)
R=0.1 (Expt.)
R=0.1 (ANN)
-0.2
-0.4
-0.6
2
3
4
5
6
7
Log fatigue life, cycles
Fig. 7 Variation of fatigue life with peak stress (HTA/919)
1.5
1
R=10(ANN)
R=-1.5 (Expt.)
0.5
R=-1.5 (ANN)
R=-1 (Expt.)
0
R=-1 (ANN)
R=-0.3 (Expt.)
R=-0.3 (ANN)
-0.5
-1
3
4
5
6
7
Log fatigue life, cycles
Fig. 8 Variation of fatigue life with peak stress for IM7/977
1.5
R=10 (Expt.)
1
R=10(ANN)
Peak stress, GPa
Peak stress, GPa
R=10 (Expt.)
R=-1.5(Expt.)
R=-1.5 (ANN)
0.5
R=-1 (Expt.)
R=-1 (ANN)
R=-0.3 (Expt.)
0
R=-0.3 (ANN)
R=0.1 (Expt.)
R=0.1(ANN)
-0.5
-1
2
3
4
5
6
7
8
Log fatigue life, cycles
Fig. 9 Variation of fatigue life with peak stress (HTA/913)
1.5
R=10 (Expt.)
Peak stress, GPa
1
R=10 (ANN)
R=-1.5 (Expt.)
R=-1.5 (ANN)
0.5
R=-1 (Expt.)
R=-1(ANN)
0
R=-0.3 (Expt.)
R=-0.3 (ANN)
Percentage error
3
R=0.1 (Expt.)
-0.5
y = 1.4582x + 1.4966
2.75
2.5
2.25
2
R=0.1 (ANN)
0.4
0.5
0.6
0.7
0.8
0.9
Strength factor
-1
3
4
5
6
7
Log fatigue life, cycles
Fig. 12 Effect of monotonic properties on predicted
fatigue lives for R = -1, peak stress = 0.5 GPa, and
probability of failure = 0.5
Fig. 10 Variation fatigue life with peak stress for
T800/924
3
y = -1.6126x + 3.4938
25
Percentage error
Percentage error
30
y = 8.9147x + 1.0324
20
15
10
2.75
2.5
2.25
5
2
0
0
0.2
0.4
0.6
0.8
Standard deviation
Fig. 11 Effect of standard deviation on predicted
fatigue life for IM7/977
1
0.5
0.6
0.7
Volume fraction
Fig. 13 Effect of volume fraction on predicted
fatigue life
From the above discussion it can be concluded that fatigue life and hence, the fatigue life prediction by ANN is
affected by variety of influential factors such as stress amplitude or peak stress, stress ratio or mean stress or
variable stress, volume fraction of fiber, mechanical properties of composite etc. Present study also
demonstrates that statistical scatter of the fatigue life also influences on the prediction of fatigue life by ANN.
Hence to obtain an optimal neural network model it should be designed and trained considering all such
influential factors including statistical scatter of fatigue life data.
Application to new materials
Once the well trained ANN have been obtained, the possibility of predicting fatigue life of new material of other
group may be analyzed. As a second step of this analysis and to study the capability of ANN method, samples
of other glass fiber composites G/913, G913/913, G913/920, G913/SiC, G913/PE, G913/G laminates were
analyzed. Some of the results are presented in figures 14.
G 9 1 3 /G
0 .8
0 .7 5
P TD
P e a k S tr e s s , G P a
0 .7
E XP T
0 .6 5
0 .6
0 .5 5
0 .5
0 .4 5
0 .4
3
3 .5
4
4 .5
5
5 .5
6
L o g fa ti g u e l i fe
G 9 1 3 /9 1 3
0 .7 5
0 .7
P E A K S T R E S S ,G P a
0 .6 5
0 .6
0 .5 5
P td
0 .5
E XP T
0 .4 5
0 .4
0
2
4
6
8
L o g F a ti g u e l i fe
0 .8
ANN
P E A K S T R E S S ,G P a
0 .7 5
E XP T
G 9 1 3 /P E
0 .7
0 .6 5
0 .6
0 .5 5
0 .5
0 .4 5
3
3 .5
4
4 .5
5
5 .5
6
L O G F A T IG U E L IF E ,C Y C L E S
Figure. 14 Prediction of fatigue life cycles of glass fiber composite materials by ANN
CONCLUSIONS
1.
2.
3.
4.
5.
Artificial neural networks (ANNs) can be used as efficient tool in predicting the fatigue life of composite
material.
Prediction of fatigue lives of composite materials is affected by various factors such as stress amplitude,
stress ratio, monotonic properties, fiber volume fraction etc. Statistical scatter of fatigue life data also
influences the fatigue life prediction.
ANN predicts results with reasonable good accuracy as more than 92 percent of the results obtained
have percentage error of less than 5 percent.
ANN can be used to predict the fatigue life at low probability level which is desirable in critical design
applications.
ANN modeled for a carbon fiber reinforced plastic (CFRP) materials cannot be used to predict results for
other materials such as glass fiber reinforced plastic (GRP) materials if both materials are not included
in the training.
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