Predicting the Fatigue Life of Different Composite

Predicting the Fatigue Life of Different CompositeMaterials Using Artificial Neural Networks

M. Al-Assadi & H. El Kadi & I. M. Deiab

Received: 28 May 2009 /Accepted: 30 June 2009 /Published online: 16 July 2009# Springer Science + Business Media B.V. 2009

Abstract Artificial Neural Networks (ANN) have been recently used in modeling themechanical behavior of fiber-reinforced composite materials including fatigue behavior.The use of ANN in predicting fatigue failure in composites would be of great value if onecould predict the failure of materials other than those used for training the network. Thiswould allow developers of new materials to estimate in advance the fatigue properties oftheir material. In this work, experimental fatigue data obtained for certain fiber-reinforcedcomposite materials is used to predict the cyclic behavior of a composite made of adifferent material. The effect of the neural network architecture and the training functionused were also investigated. In general, ANN provided accurate fatigue life prediction formaterials not used in training the network when compared to experimentally measuredresults.

Keywords Fatigue . Artificial neural networks . Fiber reinforced composite materials

1 Introduction

Polymer-matrix composites are finding increased use in aerospace, automotive, marine andcivil infrastructure applications. In many of these applications, the material used issubjected to cyclic loading triggering questions about the fatigue behavior of thesematerials. Since most of these composites are made from laminates consisting ofunidirectional laminae, predicting the fatigue behavior of these laminae could be the initialstep towards predicting the overall behavior of the laminate under cyclic loading.

Artificial Neural Networks (ANN) have proved to be useful for various engineeringapplications. Due to their massively parallel structure, ANN can deal with manymultivariable non-linear modeling for which an accurate analytical solution is difficult toobtain. ANN have already been used in medical applications, image and speechrecognition, classification and control of dynamic systems, among others; but only recently

Appl Compos Mater (2010) 17:1–14DOI 10.1007/s10443-009-9090-x

M. Al-Assadi :H. El Kadi (*) : I. M. DeiabCollege of Engineering, American University of Sharjah, Sharjah, UAEe-mail: [email protected]

have they been used in modeling the mechanical behavior of fiber-reinforced compositematerials [1, 2]. The system can be considered as a black box and it is unnecessary to knowthe details of its internal behavior. These nets may therefore offer an accurate and costeffective approach for modeling fatigue life.

Artificial neural networks have been previously used to predict the fatigue life of thesame composite material used in the training process of the ANN under different loadingconditions. El Kadi, in a recent review [1], showed that ANN can give accurate predictionsif not better than those normally obtained by conventional methods. It was also shown that,the accuracy of the network depends on the appropriate ANN architecture, the number ofhidden layers and the number of neurons in each hidden layer.

Lee et al. [3] evaluated the performance of neural networks in predicting the fatigue lifeof carbon fiber and glass fiber-reinforced composites subjected to different stress ratios. Themaximum and minimum stress and failure probability level were used as the inputparameters while the number of cycles to failure was the output from the network. Theyconcluded that ANN can be trained to model constant-stress fatigue behavior at least aswell as other current life-prediction methods.

Al-Assaf and El Kadi [4] trained a feed-forward neural network (FNN) to predict fatiguefailure of unidirectional glass/epoxy under tension-tension and tension-compressionloading. They used a unidirectional material with different fiber orientation anglessubjected to three stress ratios. The input parameters to the network were stress ratio,fibre orientation and the maximum stress while the output was the number of cycles tofailure. A back propagation training algorithm was used. In spite of the small number oftraining data points used, the ANN predictions compared well with the experimental data.In a later work, El Kadi and Al-Assaf [5] used different neural networks structures topredict fatigue life and compared the results with those previously obtained using FNN. Asbefore, the fiber orientation angle, the maximum stress and stress ratio were the inputs tothe network that will predict the fatigue life of the unidirectional glass/epoxy composite.Four ANN architectures were investigated: modular neural networks (MNN), Radial basisfunction networks (RBF), self organizing features maps (SOFM) and Principal componentanalysis (PCA). They concluded that MNN with five sub-networks gave the best results.The normalized mean square-error was reduced from 14.27% in the case of FNN to 5.7%for MNN.

Constant fatigue life diagrams are typically used to show the overall behavior of amaterial under cyclic conditions for various stress ratios. Building such a diagram requires alarge number of experimental results and the successful use of artificial neural networks inbuilding constant fatigue life diagrams for composite materials would reduce the number ofexperiments needed. Freire Jr. et al. [6] used twelve different S-N curves for a plasticreinforced with fiberglass of [90/0/±45/0]s configuration in conjunction with a multi-layerFNN trained with a back-propagation algorithm. The input parameters were the mean stressand the number of cycles while the lone output parameter was the alternating stress. Onehidden layer containing 2–30 neurons was used in the study. Training of the ANN wasattempted with data obtained from 3, 4, 5 and 6 values of the stress ratio. Even by onlyusing three S-N curves in the training, satisfactory results were obtained. On the other hand,when the number of training S-N curves increased, more reliable solution was obtained. Ina follow-up study [7], using a modular neural network in the development of the constantlife diagram, led to better results compared to the traditional feed-forward neural network.As for the FNN study, the results obtained using 6 values of the stress ratio in the trainingproduced the best results. They also concluded that increasing the number of modules givesbetter predictions only if the training set is also increased.

2 Appl Compos Mater (2010) 17:1–14

One of the anticipated benefits of the successful application of ANNs would be that itcould be possible to predict the cyclic behavior of a material for which no fatigue data isavailable by using the known characteristics of other composites. Lee et al. [3] trained anANN on fatigue data from four different material systems to predict the cyclic behavior ofan additional material not used in the training. The results obtained appear unsatisfactory asthe average root mean square error (RMSE) was of the order 100% for a material with thesame fiber system and of the order of 170% if the fiber used in the trained system is not ofthe same type used for the tested case. They consequently concluded that there seems littleprospect of transferring the predictive capability of a network with any degree of accuracyfrom one family of composites to another.

El Kadi and Al-Assaf [8], in a preliminary study, trained a MNN to predict the numberof cycles to failure for different composite materials subjected to fatigue under constantstress ratio. Seven different materials were used to train the neural network in order topredict the fatigue behavior of an eighth material. The RMSE obtained was found to be36.2%. The current investigation is continuation of this work.

2 Experimental Fatigue Data

This work addresses the behavior of unidirectional fiber reinforced composites subjected totension-tension fatigue loads. Constant stress ratio fatigue data collected from a variety ofpublished investigations [9–15] are be used to test the suitability of the artificial neural networksin predicting the fatigue behavior of a composite not used in the training of the network. Oncethe procedure has been shown to generate acceptable predictions, the same method can beextended to predict the fatigue behavior under different values of the stress ratio. Theexperimental data used here is obtained for a constant stress ratio, R ¼ smin=smax ¼ 0:1.Table 1 shows the materials and fiber orientation angles of the experimental data used.

3 Artificial Neural Networks

ANN generally consist of a number of layers (Fig. 1): the layer where the input patterns areapplied is called the input layer. This layer could include static & cyclic properties of thecomposite material under consideration, its lay-up, the maximum applied stress, the stress

Table 1 Experimental fatigue data used in the current investigation

Material Fiber Orientation angles Reference

Gevetex/Bakelite E-Glass/Epoxy 0, 5, 10, 15, 20, 30, 60 Hashin & Rotem [9]

AS/3501-5A Graphite/Epoxy 0, 10, 20,30,45,60,90 Awerbuch Hahn [10]

Scotchply 1003 Glass/Epoxy 0, 19, 45,71,90 El Kadi & Ellyin [11]

E-Glass/Polyester 0, 15, 45,75,90 Philippidis & Vassilopous [12]

T800H/2500 Carbon/Epoxy 0, 10, 15,30,45,90 Kawai & Suda [13]

DOE-MSU Glass/Polyester 0, 90 Epaarachchi& Clausen [14]

XAS/914 Carbon/Epoxy 0 Fernando & Dickson & Adam & Reiter& Harris [15]

Kevlar/914 Kevlar/Epoxy 0 Fernando & Dickson & Adam & Reiter& Harris [15]

Appl Compos Mater (2010) 17:1–14 3

ratio, etc. The layer where the output is obtained is the output layer which could, forexample, contain the fatigue life of this composite under the specific loading conditions. Inaddition, there may be one or more layers between the input and output layers called hiddenlayers, which are so named because their outputs are not directly observable. The additionof hidden layers enables the network to extract higher-order statistics which are particularlyvaluable when the size of the input layer is large [16]. Neurons in each layer are fully orpartially interconnected to preceding and subsequent layer neurons with each interconnec-tion having an associated connection strength (or weight). The input signal propagatesthrough the network in a forward direction, on a layer-by-layer basis. These networks arecommonly referred to as multilayer feed-forward neural networks (FNN). Manypublications discuss the development and theory of ANN (see for example [16–19]).

Although all neural network models share common operational features, input require-ments and modeling and generalization abilities are different. Consequently, each paradigmwould have advantages and disadvantages depending on the particular application andselecting the appropriate network class with suitable parameters is vital to ensure asuccessful application. More details about the various ANN structures, their similarities andtheir differences can be found in [16–19]. In addition to the feed-forward neural network,the following architectures will be used in this work:

Cascade—Forward Neural Networks (CFFN) CFFN are similar to feed-forward networks,but include a weight connection from the input to each layer, and from each layer to thesuccessive layers. For example, a three-layer network has connections from layer 1 to layers2, layer 2 to layer 3, and layer 1 to layer 3. The three-layer network also has connectionsfrom the input to all three layers. The additional connections might improve the speed atwhich the network learns the desired relationship.

Elman Networks (ELM) ELM are multi-layer back-propagation networks, with the additionof a feedback connection from the output of the hidden layer to its input. This recurrentconnection allows ELM networks to learn to recognize and generate temporal patterns, aswell as spatial patterns [19]. A two-layer Elman network is shown in Fig. 2 [20].

Fig. 1 General configuration of a feed-forward artificial neural network (with permission of Elsevier) [1]


Layer Recurrent Network (LRN) LRN distinguishes itself from a feed-forward neuralnetwork in that it has at least one feedback loop. For example a LRN may consist of asingle layer of neurons with each neuron feeding its output signal back to the inputs of allthe other neurons. The presence of feedback loops has a profound impact on the learningcapabilities of the network and its performance. Moreover, the feedback loops involve theuse of particular branches composed of unit-delay elements resulting in a nonlinearbehavior assuming that the neural network contains nonlinear units [17]. Since fatiguepredictions are not a temporal process, one would not see the advantage of using LRN topredict fatigue life. However, benefits may result due to having a feedback-structure patternof the network that the overall weights calculations may result in obtaining a more optimalstatic neural network.

4 ANN Training Algorithms

The back-propagation training algorithm [18] is commonly used to iteratively minimizethe following cost function with respect to the interconnection weights and neuronsthresholds:

E ¼ 1

2

XP

1

XN

i¼1

ðdi � OiÞ2 ð1Þ

where P is the number of experimental data pairs used in training the network and N is thenumber of output parameters expected from the ANN. di and Oi could be the experimentalnumber of cycles to failure and the current life prediction of the ANN for each loadingcondition i respectively. Iteratively, the interconnection weights between the jth node andthe ith node are updated as:

wji t þ 1ð Þ ¼ a wjiðtÞ þ h xi f0 netkj

� � XN

l¼1

dl � Olð Þ f 0 net0l� �

wlj ð2Þ

where α is a momentum constant, η the learning rate, xi the input pattern at the iterativesample t, net0N the input to node N at the output layer, netkj is the input to a node j in the kth

layer and the function f′ is the derivative of the neuron activation function. The learning

Input vectorp(x)=[1 x1.....xn]

input layer weights Hidden LayerTansig functions

Output layerWeights

Output layerPurelinefunction

Recurrentweights

Fig. 2 A two-layer recurrent Elman neural network architecture (with permission of Elsevier) [20]


rate determines what amount of the calculated error sensitivity to weight change will beused for the weight correction. It affects the convergence speed and the stability of weightsduring learning. The “best” value of the learning rate depends on the characteristics of theerror surface. For rapidly changing surfaces, a smaller rate is desirable while for smoothsurfaces, a larger value of the learning rate will speed up convergence. The momentumconstant (usually between 0.1 and 1) smoothes weight updating and prevents oscillationsin the system and helps the system escape local minima in the training process by makingthe system less sensitive to local changes. Much as the learning rate, the momentumconstant “best” value is also peculiar to specific error surface contours. The trainingprocess is terminated either when the Mean-Square-Error (MSE), Root-Mean-Square-Error(RMSE), or Normalized-Mean-Square-Error (NMSE), between the actual experimentalresults and the ANN predictions obtained for all elements in the training set has reached apre-specified threshold or after the completion of a pre-specified number of learningepochs.

In addition to the typical back-propagation algorithm, the following training functionsare also considered in this study:

Resilient Back-Propagation (RP) Multilayer networks typically use sigmoid transferfunctions in the hidden layers. These functions are often called “squashing” functions,because they compress an infinite input range into a finite output range. Sigmoidfunctions are characterized by the fact that their slopes must approach zero as the inputgets large. This causes a problem when steepest descent is used to train a multilayernetwork with sigmoid functions because the gradient can have a very small magnitudeand, therefore, cause small changes in the weights and biases, even though the weightsand biases are far from their optimal values. The purpose of the resilient back-propagationtraining algorithm is to eliminate these harmful effects of the magnitudes of the partialderivatives [21].

Gradient Descent (GD) In the steepest descent training function, the weights and biases areupdated in the direction of the negative gradient of the performance function. The learningrate is multiplied by the negative of the gradient to determine the changes to the weightsand biases. The larger the learning rate, the bigger the step. If the learning rate is made toolarge, the algorithm becomes unstable. If the learning rate is set too small, the algorithmtakes a long time to converge. The training stops if the number of iterations exceeds thepredetermined number of epochs, the performance function drops below a specific goal, themagnitude of the gradient is less than a stipulated value, or the training time surpasses apreset time [21].

Gradient Descent with Momentum (GDM) Gradient descent with momentum allows anetwork to respond not only to the local gradient, but also to recent trends in the errorsurface. Acting like a low pass filter, momentum allows the network to ignore smallfeatures in the error surface. Without momentum a network can get stuck in a shallow localminimum. With momentum a network can slide through such a minimum [21].

Variable Learning Rate (GDA) With standard steepest descent, the learning rate is heldconstant throughout training. The performance of the algorithm is very sensitive to theproper setting of the learning rate. If the learning rate is set too high, the algorithm canoscillate and become unstable. If the learning rate is too small, the algorithm takes too longto converge. It is not practical to determine the optimal setting for the learning rate before


training, and, in fact, the optimal learning rate changes during the training process, as thealgorithm moves across the performance surface. The performance of the steepest descentalgorithm can be improved by allowing the learning rate to change during the trainingprocess. An adaptive learning rate attempts to keep the learning step size as large aspossible while keeping learning stable. The learning rate is made responsive to thecomplexity of the local error surface [21].

Variable Learning Rate with Momentum (GDX) This function combines adaptive learningrate with momentum training. It is invoked in the same way as GDA except that it has themomentum coefficient as an additional training parameter [21].

5 Predicting Fatigue Life Using ANN

In this study, the input parameters to the ANN were comprised of a combination of thefollowing monotonic and cyclic properties: the modulus of elasticity in the direction of thefiber (E0), the modulus of elasticity in the direction perpendicular to the fibers (E90),the tensile strength of the laminate in the fiber direction (S0T), the tensile strength of thelaminate in the direction perpendicular to the fibers (S90T), the fiber orientation angle (θ)and the maximum applied stress (σmax). The number of cycles to failure (Nf) was the soleoutput from the network.

Since the range of number of cycles to failure varied between 10 and 8,000,000cycles, training the networks to learn such a wide range will produce unacceptable andunbalanced modeling performance. This will occur since the ANN will strive tominimize the overall error for all input patterns. Hence, minimizing the differencebetween the network output and observed data for high values of stress cycles wouldlead to incorrect results for the patterns associated with lower values of number of cyclesto failure. A more suitable method would be the normalization of the logarithmic valuesof the number of cycles to reach a range between 0 and 1. The maximum applied stressvaried between 12 to 1900 MPa. These values were also normalized after taking thelogarithmic values of the stress reducing the scale to values between 0 and 1. All othermechanical properties as well as fiber orientation angles were normalized linearlybetween 0 and 1 in the usual fashion. The Matlab Software [21] was used to construct,train and test the networks.

Fatigue experimental data from the materials shown in Table 1 was used to train and testthe network. The network was trained using all-but-one of the materials while the testingwas done for the remaining material. The effects of ANN architecture, training algorithm aswell as number of neurons per hidden layer were considered to obtain the optimum fatiguelife prediction.

5.1 Effect of Training Functions

The training functions introduced in section 4, were used to predict the fatigue life of thevarious materials. Table 2 shows a typical comparison between the RMSE obtained with thevarious training functions using 16 and 20 neurons for Scotchply 1003 Glass/Epoxylaminae with 0°, 19°, 45°, 71° and 90° fiber orientations. As shown, resilient back-propagation resulted in the lowest RMSE in all cases considered. Similar results wereobtained for other materials [22]. Therefore, throughout the rest of this study, only resultsobtained using this training function will be reported.


5.2 Effect of Number of Hidden Neurons

Using feed-forward neural networks with resilient back-propagation training, the effect ofvarying the number of hidden neurons on the fatigue life prediction was investigated. Thenumber of neurons in the hidden layer was varied to obtain the lowest RMSE. Figure 3shows typical variations of RMSE with the number of hidden neurons obtained whilepredicting fatigue life of Scotchply 1003 glass/epoxy, AS/3501-5A graphite/epoxy andT800H/2500 carbon/epoxy.

0

5

10

15

20

25

30

0 5 10 15 20 25 30

RM

SE

Number of Hidden Neurons

Scotchply 1003 Glass/Epoxy

AS/3501-5A Graphite/Epoxy

T800H/2500 Carbon/Epoxy

Fig. 3 Effect of the number of hidden neurons on the RMSE for different materials

Table 2 RMSE obtained as a function of the type of architecture and training function

Training Function Neural Network Architecture

FFN CFFN ELM

16neurons

20neurons

16neurons

20neurons

16neurons

20neurons

Resilient Back-propagation (RP) 15.60% 9.70% 16.80% 21.10% 13.60% 15.60%

Gradient descent (GD) 53.50% 23.10% 34.50% 40% 17.62% 35.10%

Gradient descent with momentum (GDM) 15.70% 17.80% 35.40% 26.80% 19.50% 25.20%

Variable Learning Rate (GDA) 33.10% 19.10% 22.20% 23.10% 34.60% 17.02%

Variable Learning Rate with momentum(GDX)

17.60% 24.30% 23.90% 30.60% 30.40% 18.70%


5.3 Effect of Network Architecture

Using resilient back-propagation training, different network architectures are used to predictthe fatigue life of some of the materials under consideration. As mentioned before, in eachcase all-but-one of the materials will be used to train the neural network and the remainingmaterial will be used to test the network.

Figures 4, 5, 6 and 7 show typical fatigue life predictions of AS/3501-5A Graphite/Epoxy using different neural network architectures with one hidden layer containing 16–20neurons. For comparison purpose, resilient back-propagation is used in all cases. TheRMSE obtained using feed-forward, cascade-forward, Elman neural and layer recurrentnetworks were found to be 12.3%, 8.8%, 9.2% and 13.25% respectively. The figures showthat the resilient backpropagation cascade-forward architecture predicts fatigue life with theleast error. The figures also show a shift between the experiments and the predicted valuesalong the Nf axis for the laminae with zero fiber orientations. This might be due to thesignificantly different failure modes of these laminae compared to all other off-axisspecimens.

A similar investigation was conducted to predict the fatigue life of othercomposites. Figures 8, 9 and 10 show typical fatigue life predictions of Scotchply1003 glass/epoxy obtained using different network architectures and number of hiddenneurons. The best predictions in this case were obtained using a layer recurrent neuralnetwork with one hidden layer and 16 hidden neurons (RMSE=11.5%). In this case, theabove-mentioned shift observed along the Nf axis is not as prevalent as for the case ofGraphite/Epoxy.

Figure 11 shows an example of the fatigue life prediction of T800H/2500 carbon/epoxyusing a feed-forward back-propagation network with one hidden layer and 20 hidden

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Fig. 4 Fatigue life prediction of AS/3501-5A Graphite/Epoxy using FNN with 20 neurons


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Fig. 6 Fatigue life prediction of AS/3501-5A Graphite/Epoxy Using ELM with 20 neurons

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Fig. 5 Fatigue life prediction of AS/3501-5A Graphite/Epoxy Using CFFN with 20 neurons


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Fig. 8 Fatigue life prediction of Scotchply 1003 Glass/Epoxy Using ELM with 16 neurons

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Fig. 7 Fatigue life prediction of AS/3501-5A Graphite/Epoxy Using LRN with 16 neurons


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Fig. 10 Fatigue life prediction of Scotchply 1003 Glass/Epoxy Using LRN with 16 neurons

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Fig. 9 Fatigue life prediction of Scotchply 1003 Glass/Epoxy Using ELM with 20 neurons


neurons. Although the RMSE obtained in this case was 16.3%, the overall prediction of theexperimental data is not as accurate as in the previous two cases. This could be attributed tothe lowest number of experimental points available for this material compared to theprevious two cases. With the larger number of experiments used in the training, this couldcause the ANN to, not only predict the trend of the fatigue behavior, but also the manyvariations within the experimental data used in training. This could be negatively affectingthe prediction capability of the network.

The predictions obtained show that, although acceptable fatigue life predictions of newmaterials can be obtained using ANN, there is no specific network architecture/trainingalgorithm combination that always results in the best fatigue life prediction for all materials.Additional investigation is needed to determine the possibility of finding uniquecombinations that always result in the best fatigue life predictions.

6 Conclusion

Different neural network architectures using a variety of training functions were used topredict the fatigue life of fiber reinforced composite materials. Training was performed oncertain composites while the prediction was done for different materials. The results showthat ANN can be accurately used to predict the fatigue failure of a composite material notused in the training of the network. The results however show that no singulararchitecture/training method combination can always produce the best results for allmaterials.

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Fig. 11 Fatigue life prediction of T800H/2500 Carbon/Epoxy Using FNN with 20 neurons


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