Using Artificial Neural Networks to Predict the Fatigue

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Using Artificial Neural Networks to Predict the Fatigue

Life of Different Composite Materials Includingthe Stress Ratio Effect

Mohamed Al-Assadi & Hany A. El Kadi &

Ibrahim M. Deiab

Received: 14 June 2010 / Accepted: 4 August 2010 / Published online: 13 August 2010# Springer Science+Business Media B.V. 2010

Abstract Artificial Neural Networks (ANN) have been successfully used in predicting the

fatigue behavior of fiber-reinforced composite materials. In most cases, the predictions were

obtained for the same material used in training subjected to different loading conditions.

The method would be of greater value if one could predict the failure of materials other

than those used for training the network. In a recent paper, ANN trained using the

experimental fatigue data obtained for composites subjected to a constant stress ratio

R ¼ s min=s maxð Þ was successfully used to predict the cyclic behavior of a composite madeof a different material. In this work, this method is extended to include the stress ratio

effect. The results show that ANN can provide accurate fatigue life prediction for different

materials under different values of the stress ratio. These results can allow for the

development of a materials smart database that can be used for various engineering

applications.

Keywords Artificial neural networks . Composite materials . Fatigue . Stress ratio

1 Introduction

In many applications using polymeric composites, the material is subjected to cyclic

loading. It is therefore important to be able to accurately predict the behavior of these

materials under such conditions. With the unidirectional lamina being the building block of

the laminate, it might also be beneficial to initially predict the behavior of the lamina under

cyclic loading. These results can hopefully be extended to predict the fatigue behavior of

laminates.

Artificial Neural Networks (ANN), with their massively parallel structure, can deal with

many multivariable non-linear problems for which an accurate analytical solution isdifficult to obtain. In the area of composite materials, ANN have already been used in

modeling the static and cyclic behavior of these materials, controlling the manufacturing

Appl Compos Mater (2011) 18:297–309

DOI 10.1007/s10443-010-9158-7

M. Al-Assadi : H. A. El Kadi (*) : I. M. Deiab

College of Engineering, American University of Sharjah, Sharjah, United Arab Emirates

e-mail: [email protected]



parameters used in their machining, predicting their wear properties, among others [1, 2].

The system can be considered as a black box rendering the knowledge of its internal

behavior unnecessary to most users.

ANN generally consist of an input layer (where the input patterns are applied), an output

layer (where the output is obtained) and one or more hidden layers (so named because their outputs are not directly observable) between the input and output layers (see Fig. 1) [1]. For

predicting the fatigue life of composites, the input parameters may include static and cyclic

properties of the composite material under consideration, its lay-up, the maximum applied

stress, the stress ratio, etc. The output parameter could be the fatigue life of this composite

under the given loading conditions. Neurons in each layer are fully or partially

interconnected to preceding and subsequent layer neurons with each interconnection

having an associated connection weight. The input signal propagates through the network

in a forward direction, on a layer-by-layer basis. These networks are commonly referred to

as multilayer feed-forward neural networks (FNN). Many publications discuss the

development and theory of ANN [for example, 3–7].

Even though all neural network architectures share common operational features, input

requirements and modeling and generalization abilities could be different. Consequently,

each structure would have its pros and cons depending on the particular application it is

used for and selecting the appropriate network class is imperative to ensure accurate results.

More details about the various ANN structures, their similarities and differences can also be

found in [3–7].

The back-propagation training algorithm [5] is commonly used to iteratively minimize

the following cost function with respect to the interconnection weights:

E ¼1

2

X P

1

X N

i¼1

d i À Oið Þ2

where P is the number of experimental data pairs used in training the network and N is the

number of output parameters expected from the ANN. d i and Oi represent the experimental

number of cycles to failure and the current life prediction of the ANN for each loading

condition i respectively.

Fig. 1 General configuration of an artificial neural network (with permission of Elsevier) [ 1]

298 Appl Compos Mater (2011) 18:297–309



The training process is terminated either when the Mean-Square-Error (MSE), Root-

Mean-Square-Error (RMSE), or Normalized-Mean-Square-Error (NMSE), between the

actual experimental results and the ANN predictions obtained for all elements in the

training set has reached a pre-specified threshold or after the completion of a pre-specified

number of learning epochs.

2 Fatigue Failure Prediction of Composites Using ANN

Artificial neural networks have been successfully used to predict the fatigue life of a

composite material subjected to different loading conditions. El Kadi, in a recent review [1],

showed that ANN can give comparable predictions to those normally obtained by

conventional methods. It was also shown that factors such as ANN architecture, input

parameters, number of hidden layers and number of neurons in each hidden layer, can

influence the accuracy of the results obtained.

The use of ANN to predict fatigue strength of APC-2 graphite/PEEK composites for 0.1

stress ratio was addressed in the work by Aymerich and Serra [8]. The input parameters to

the ANN were the number of cycles to failure and the stacking sequence of the laminate

while the fatigue strength was the sole output. The number of neurons used in the hidden

layer varied from 4 to 12 to assure a good compromise between speed and precision. The

authors concluded that ANN are able to predict fatigue life of fiber reinforced laminates

provided that a sufficiently large set of experimental data, representative of the

characteristic damage models of the category of examined sequence, is available.

Lee et al. [9] utilized ANN in predicting fatigue failure of carbon fiber-reinforcedlaminates under various stress ratios (0.1 to 10). They investigated the use of various input

parameters to find the combination resulting in the optimum fatigue life prediction. Using

the maximum and minimum values of the stress as well as the failure probability level as

input parameters and the number of cycles to failure as the output resulted in the best

predictions. The authors also investigated the effect of the number of hidden layers and the

number of stress ratios used in training on the fatigue life prediction accuracy. Their results

showed that ANN can be trained to model constant-stress fatigue behavior at least as well

as other current life-prediction methods.

The use of ANN to predict the fatigue failure of unidirectional glass/epoxy composite for

a range of fiber orientation angles under various loading conditions was also considered byAl-Assaf and El Kadi [10]. Feed-forward neural networks (FNN) provided accurate

relationship between the input parameters (maximum stress, stress ratio, fiber orientation

angle) and the number of cycles to failure. In spite of the small number of training data

points used, the results obtained were found to be comparable to other current fatigue life-

prediction methods. In a quest to improve the fatigue-life prediction accuracy, other types of

ANN structures were used [11]. Radial Basis Function (RBF), Modular (MNN), Self-

Organizing Features Maps (SOFM) and Principal Component Analysis (PCA) neural

networks were considered and their predictions compared to achieve the above-mentioned

objective. The modular networks resulted in the most accurate prediction of the fatigue lifefor the material under consideration as the normalized mean square-error was reduced from

14.27% in the case of FNN to 5.7% for MNN.

Vassilopoulos et al. [12] demonstrated that ANN is a good tool for modeling the fatigue

life of multidirectional glass fiber reinforced plastics (GFRP) composite laminates. Tension-

tension, compression-compression and tension-compression loading patterns were investi-

gated and modeling accuracy of the proposed ANN model was validated. The fiber

Appl Compos Mater (2011) 18:297–309 299



orientation, stress ratio, maximum applied stress as well as stress amplitude were the input

parameter and the number of cycles to fatigue was the output of the neural network. The

smallest values for the mean square error (MSE) were obtained with 8 hidden neurons for

the case of [0/(±45)2/0]T laminates and with 24 hidden neurons for the case of [90/0/±45/0] s

laminates. Comparing the predictions with the experimental results showed a goodagreement.

Freire Júnior et al. [13] used a modular network to predict the fatigue life of [90/0/±45/0]8fiberglass laminates tested under 12 values of the stress ratio. One hidden layer with a number

of neurons ranging between 4 and 30 was used. The input parameters to the network were the

alternating stress and the number of cycles to fatigue and the output was the mean stress.

Training of the ANN was attempted with data obtained from 3 to 6 values of the stress ratio.

Even by only using three S-N curves in the training, satisfactory results were obtained. A

more reliable solution was obtained when the number of training S-N curves increased. The

modular neural network used gave better predictions than the authors had previously obtained

using feed forward neural networks [14]. Freire Júnior et al. [15] later assessed the

applicability of two ANN architectures (multi-layer feed-forward and modular) in the

prediction of fatigue life in composites compared to the equation developed by Adam [16] for

modeling the constant-life diagram. GFRP in the form of laminar structures with distinct

stacking sequences were used in the study. These materials were tested for six different stress

ratios. The results showed that modeling of constant-life diagram can be done using ANN

while requiring a much smaller set of experimental data compared to Adam’s equation.

Analysis of the model created with the modular network architecture showed that this

network produced much better results than those obtained by both the feed-forward network

and by Adam’s equation for all the laminates analyzed.

In all the previously-mentioned studies using ANN to predict the fatigue life of fiber

reinforced composites, the authors only used one specific material in their study. It should

be mentioned however, that one of the anticipated benefits of the successful application of

ANN should be the possibility of predicting the lives of materials for which no fatigue data

is available by using known cyclic characteristics of other materials.

3 Fatigue Failure Prediction of New Composite Materials

Lee et al. [9] trained an ANN on data from four different material systems to predict thefatigue properties of a fifth material not used in the training. Monotonic mechanical

properties of this additional material were also used in training. They obtained predictions

with a root mean square error (RMSE) of the order of 100% for the prediction of the same

material system and of 170% for a different material systems (for example, carbon fiber

systems in training vs. glass fiber system in testing). The authors therefore concluded that it

is unlikely to transfer the accurate predictive capability of a network from one family of

composites to another. El Kadi [1] had however suggested that better predictions might be

achieved if a larger number of representative materials was used in the testing and

appropriate material properties were used in both the training and the testing stages.El Kadi and Al-Assaf [17], in a preliminary study, trained a modular neural network to

predict the number of cycles to failure for different composite materials under a constant

stress ratio. They used five materials to train the ANN and one additional material for

testing. The input parameters were comprised of monotonic and cyclic properties (strength,

modulus, fiber orientation, maximum applied stress). The RMSE was found to be 36.2%

well below the value obtained by Lee et al. [9].




In a recent study, Al-Assadi et al. [18] used ANN to predict the fatigue life for composite

materials other that those used for training. Seven different materials were used in training

the network for the purpose of predicting the fatigue behavior of an eighth material. All

materials used were subjected to a cyclic loading with a constant stress ratio of 0.1. The

effect of the ANN architecture, the type of training algorithm as well as the number of neurons per hidden layer was addressed. Although no singular ANN architecture/training

method combination was shown to consistently produce the best predictions for all

materials, the results showed that ANN can accurately predict the fatigue life of a

composite not used in the training of the network. Depending on the material and the

network architecture used, the typical root mean square error obtained varied between 8.8%

and 16.3%. These predictions compared well with those obtained by Lee et al. [9] where

the average root mean square error was much higher as previously mentioned in this

section. The current investigation is an extension of this work for fatigue data taking the

effect of the stress ratio into consideration.

4 Experimental Fatigue Data

The current work addresses the fatigue behavior of unidirectional fiber reinforced laminates

subjected to tension-tension, tension-compression and compression-compression fatigue

loads. Experimental fatigue data collected from a variety of published works [19–27] was

used to train and test the ANN. This data included results obtained for various types of

composite materials with numerous fiber orientation angles subjected to several values of

the stress ratio. Table 1 shows the source of the experimental fatigue data used in thecurrent investigation.

5 Artificial Neural Networks

Different neural network architectures with a variety of training algorithms are used in this

work to predict the fatigue life of fiber reinforced composite materials under a range of

Table 1 Source of experimental fatigue data used in the current investigation

Material Fiber orientation angles Stress ratio Reference

Glass/Epoxy 0,5,10,15,20,30,60 0.1 Hashin & Rotem [19]

AS/3501-5A Graphite/

Epoxy

0,10,20,30,45,60,90 0.1 Awerbuch & Hahn [20]

Scotchply 1003 Glass/Epoxy 0,19,71,90 0.1,0.5,−1 El Kadi & Ellyin [21]

E-Glass/Polyester 0,15,30,45,60,90 0.1,0.5,−1,10 Philippidis & Vassilopoulos

[22]

T800H/2500 Carbon/Epoxy 0,10,15,30,45,90 0.5,0.1,−0.3,−1 Kawai & Suda [23]

APC-2 AS4 Carbon/Peek 0,15,30,60,75,90 0,0.2,5,∞ Jen & Lee [24]

Glass/Polyester 0,90 0.1,0.5,−1,2,10 Epaarachchi & Clausen [25]

XAS/914 Carbon/Epoxy 0 0.1,−0.6 Fernando & Dickson & Adam

& Reiter & Harris[26]

Kevlar /914 Carbon/Epoxy 0 0.01,0.1,−0.3,−0.6 Fernando & Dickson & Adam

& Reiter & Harris[27]




stress ratios. In addition to the feed-forward neural network, three additional network

architectures are used in the current investigation: Cascade-forward neural networks

(CFNN), Elman neural networks (ELM) and Layer recurrent neural networks (LRN). A

brief definition of these networks is shown in [1].

Besides the typical back-propagation algorithm, the following training algorithmswere also considered in this study: Levenberg-Marquardt (LM), BFGS Quasi-Newton

(BFG), Resilient back propagation (RP), Scaled conjugate gradient (SCG), Conjugate

gradient with Powell/Beale restarts (CGB) and Polak-Ribiére conjugate gradient (CGP).

Details about the properties of these training algorithms can be found in [7]. For most

of the cases considered, the RP training algorithm resulted in the lowest RMSE. For

example, Table 2 shows the RMSE obtained when predicting the fatigue life of Kevlar/

Epoxy 914 using FNN with the aforementioned training algorithms. Therefore,

throughout the rest of this study, only resilient back propagation training will be used.

The advantage of the resilient back-propagation training algorithm is to eliminate the

harmful effects resulting from using sigmoid transfer functions in the hidden layers.

Sigmoid functions are characterized by the fact that their slopes must approach zero as the

input gets large. This causes a problem when steepest descent is used to train a multilayer

network because the gradient can have a very small magnitude and, therefore, causes

small changes in the weights and biases, even though the weights and biases are far from

their optimal values.

Experimental data for the materials shown in Table 1 with the Matlab software [28]

were used to construct, train and test the networks. The effects of ANN architecture, as

well as number of hidden neurons were investigated to obtain the optimum fatigue life

prediction.

6 Predicting Fatigue Life Using ANN

In this study, the neural network is trained using all-but-one of the materials while the

testing is done for the remaining material. The input parameters to the ANN consist of

a combination of the following monotonic and cyclic properties: the modulus of

elasticity in the fibers direction (E0), the modulus of elasticity in the direction

perpendicular to the fibers (E90), the static tensile strength of the laminate in the fibers

direction S0TÀ Á

, the static tensile strength of the laminate in the direction perpendicular tothe fibers S90

TÀ Á

, the static compressive strength of the laminate in the fiber direction

S0C

À Á, the static compressive strength of the laminate in the direction perpendicular to the

fibers S90C

À Á, the fiber volume fraction (Vf ), the fiber orientation angle (θ), the maximum

Table 2 RMSE for different training algorithms for Kevlar/Epoxy 914

Number Algorithm Function name RMSE (%)

1 Trainlm (LM) Levenberg-Marquardt 14.92 Trainbfg (BFG) BFGS Quasi-Newton 117.7

3 Trainrp (RP) Resilient Back propagation 9.9

4 Trainscg (SCG) Scaled Conjugate Gradient 39.1

5 Traincgb (CGB) Conjugate Gradient with Powell/Beale Restarts 28.7

6 Traincgp (CGP) Polak-Ribiére Conjugate Gradient 26.1




applied stress (σ max), and the minimum applied stress (σ min). Although ten parameters

are needed as an input to the network, it should be emphasized that the majority of

these properties can be obtained from a few simple tension and compression tests. The

number of cycles to failure (Nf ) was the sole output from the network. It should be noted

that, in this study, since the composite might be subjected to compressive loads,compressive laminate properties had to be included as input parameters to the network.

This was not the case in [18] when the material was only subjected to a tension-tension

fatigue load.

Since the range of fatigue life varied between 3 and 10,000,000 cycles, training the

networks to learn such a wide range will produce unacceptable and unbalanced modeling

performance. This will occur since the ANN will strive to minimize the overall error for

all input patterns. Hence, minimizing the difference between the network output and

observed data for high values of stress cycles would be at the expense of the lower range

values. Classical normalization, where the range is scaled between 0 and 1, will also not

solve the problem since smaller values of life cycles will be very close to zero. To make

the output amenable for successful learning, the logarithmic values for the stress cycles

were normalized reducing the scale to lie between 0.45 and 7. The maximum applied

stress (varying between −280 and 2000 MPa) and the minimum applied stress (varying

between −840 and 1000 MPa) were normalized between −1 and 1 while the fiber

orientation angles which vary between 0° and 90° were normalized between 0 and 1 for

network training and testing.

In all cases, one hidden layer was used. Figures 2 and 3 show typical variations of

the RMSE obtained using FNN for two of the materials considered as a function of the

number of hidden neurons. For all cases considered [29], a number of hidden neurons between 6 and 12 lead to the most accurate fatigue life prediction.

Figures 4, 5, 6, 7, 8, 9 and 10 show the relation between the maximum applied stress and

the number of cycles to failure for some of the materials considered. As previously

indicated, for each of the cases, the fatigue data of the material shown was not used in

training the neural network. These figures show the experimental results as well as typical

predictions obtained using ANN. The neural network predictions were obtained using a

variety of architectures with different numbers of hidden neurons. The predictions obtained

are the average prediction of three runs with the same input parameters. This is done to

Fig. 2 Variation of RMSE with number of hidden neurons for Glass/Polyester composite




reduce the inconsistencies that can result due to the randomness of the ANN initial guess.

Using the average of a larger number of runs was not shown to result in a significant

difference in the predictions. The results show that, in general, ANN can accurately predict

the cyclic behavior of these materials.

Figures 4 and 5 show the experimental vs the predicted data for Scotchply 1003

glass/epoxy with various fiber orientation angles for two different numbers of hidden

neurons. These predictions were obtained using a FNN with a resilient back

propagation training algorithm. The resulting RMSE was reduced from 16.7% using

Fig. 4 Fatigue life prediction of Scotchply 1003 Glass/Epoxy with different fiber orientation angles using

FNN with 12 neurons

Fig. 3 Variation of RMSE with number of hidden neurons for APC-2 Carbon/PEEK composite




10 neurons to 15.9% using 12. Figures 6 and 7 show the on-axis fatigue life predictionfor XAS-914 Carbon/Epoxy using an artificial neural network with 11 neurons. In this

case, results obtained using FNN and CFNN architectures are shown. The RMSE were

found to be 13.6% and 6.1% for the FNN and CFNN respectively. Figures 8 and 9

Fig. 6 On-axis fatigue life prediction of XAS-914 Carbon/Epoxy using FNN with 11 neurons

Fig. 5 Fatigue life prediction of Scotchply 1003 Glass/Epoxy with different fiber orientation angles using

FNN with 10 neurons




show the fatigue life prediction of AS-3501-5A Graphite/Epoxy with various fiber

orientation angles using two neural networks architectures (CFNN and LRN) with 11

hidden neurons. The RMSE was reduced from 8.2% with the LRN to 7.8% usingCFNN. Figure 10 shows the experimental results of APC-2 compared to the predictions

Fig. 8 Fatigue life prediction of AS-3501-5A Graphite/Epoxy with different fiber orientation angles using

CFNN with 11 neurons

Fig. 7 On-axis fatigue life prediction of XAS-914 Carbon/Epoxy using CFNN with 11 neurons




obtained for different fiber orientation angles using a FNN with 12 neurons. For the

case shown, the RMSE obtained was found to be 13%.

It is worth mentioning here that since it is well recognized that neural networks cannot

extrapolate outside the range for which they are trained (i.e. even a well-trained ANN can

Fig. 10 Fatigue life prediction of APC-2 Carbon/PEEK with different fiber orientation angles using FNN

with 12 neurons

Fig. 9 Fatigue life prediction of AS-3501-5A Graphite/Epoxy with different fiber orientation angles using

LRN with 11 neurons




only be used to predict new data from the same knowledge domain), it is imperative to only

use ANN to predict the fatigue life of materials for which the properties fall within the

range of properties of materials used in training.

For the various cases considered, it is shown that the ANN suitably predicts the

behavior of the various materials under a variety of stress ratios. Although the ANNresults vary based on the network architecture and the training algorithm used, the

predictions obtained are acceptable in most cases. This study only addressed the fatigue

life prediction of unidirectional composite laminates; the next step is to predict the

cyclic behavior for multi-directional laminates taking into consideration the effect of the

stacking sequence.

7 Conclusions

Different neural network architectures with a variety of training algorithms were used to

predict the fatigue life of fiber reinforced composite materials under a range of stress

ratios. Training was performed on certain composites while the prediction was for

different materials not used in the training process. Depending on the material and the

network architecture used, the typical root mean square error obtained varied between

6.1% and 40%. These predictions compare well with those published in the literature

where the average root mean square error was much higher. Resilient back propagation

was found to consistently produce the best fatigue life prediction irrespective of the

type of material or the type of network architecture used. With the use of a single

hidden layer, a number of hidden neurons ranging between 6 and 12 yielded the best fatigue life predictions. Although acceptable predictions were obtained using FNN and

CFNN, more accurate results could be obtained using other ANN architectures at the

expense of a much longer training time.

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Using Artificial Neural Networks to Predict the Fatigue