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Neural network based prediction on mechanical and wear properties of short fibers reinforced polyamide composites
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Materials & Design
Materials and Design 29 (2008) 628–637 www.elsevier.com/locate/matdes
Neural network based prediction on mechanical and wear properties of short fibers reinforced polyamide composites Zhenyu Jiang a, Lada Gyurova a, Zhong Zhang a
b,*
, Klaus Friedrich a, Alois K. Schlarb
a
Institute for Composite Materials (Institut fuer Verbundwerkstoffe), University of Kaiserslautern, 67663 Kaiserslautern, Germany b National Center for Nanoscience and Technology, No. 2, 1st North Street, Zhongguancun, 100080 Beijing, China Received 11 September 2006; accepted 14 February 2007 Available online 2 March 2007
Abstract The artificial neural network technique was applied to predict the mechanical and wear properties of short fiber reinforced polyamide (PA) composites. Two experimental databases were used to train the neural network: one consisted of 101 independent fretting wear tests of PA 4.6 composites; the other one was from a commercial company and included 93 pairs of independent Izod impact, tension and bending tests of PA 6.6 composites. The predicted property profiles as a function of short fiber content or testing conditions proved a remarkable capability of well-optimized neural networks for modeling concern. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Polymer matrix; Mechanical and wear properties; Short fiber; Artificial neural network
1. Introduction Polyamide (PA) is one kind of the most versatile engineering thermoplastics, for their outstanding performance and ease of processing. It is well known that with the reinforcement of short fibers the mechanical strength and wear resistance of the polyamide composites can be significantly improved. This makes short fiber reinforced polyamide composites (SFRPA) a superior candidate for replacement of metal materials in various cases. Therefore, the design of SFRPA to meet the desired target is of importance for the present engineering application. However, the mechanical behavior and wear characteristics of SFRPA composites are currently regarded as a complicated problem, which is associated with multifold factors, e.g. composition, working condition, and manufacturing process. The property investigation of SFRPA composites is difficult to analyze theoretically, and remains to be a labor intensive and time-consuming problem for experimental research.
*
Corresponding author. Tel.: +86 10 62652699; fax: +86 10 62650450. E-mail address: [email protected] (Z. Zhang).
0261-3069/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2007.02.008
In recent years, the application of artificial neural network (ANN) has attracted extensive interests in diverse fields. Inspired by its biological counterpart, ANN uses the inter-connected mathematical nodes to imitate some abilities of human brain. Such a network can learn (or be trained) from examples, and explore by itself the underlying functional relationships between the given reasons and results. This remarkable capability of modeling is useful in the study of complicated problems, which usually cannot be solved by existing physical theories or the other mathematical approaches. Especially, the ability to find the correlation among multiple parameters with noise and to implement excellent non-linear interpolation are extremely required by industrial community for the estimation of performance of the materials with new combination of composition or processing based on the existing experimental databases. Therefore, the ANN technique has so far taken a great deal of attention in the research on mechanical and wear behaviors of polymer composites (cf. [1–3] for reviews). In the present work, the ANN with back-propagation (BP) algorithm is applied to predict the mechanical or wear properties of two kinds of SFRPA composites respectively:
Z. Jiang et al. / Materials and Design 29 (2008) 628–637
PA 6.6 and PA 4.6 composites, both of which are reinforced by short glass or carbon fibers. Based on these databases, the optimization of the neural networks is discussed through a series of comparisons. Some practical rules have been summarized. Finally the well-optimized and trained neural networks are used to predict the profile of mechanical or wear properties as a function of the content of short fiber/filler or testing condition, according to the newly constructed input data sets. 2. Neural network approach An ANN structure is commonly divided into three parts: input layer, hidden layer and output layer, as illustrated in Fig. 1. The nodes (the basic processing units, also called neurons) are connected by weighted inter-connections, which resemble the intensity of the bioelectricity transferring among the neuron cells in a real neural network. The learned knowledge can be memorized in terms of the state of these weights as well as the biases. The numbers of neurons in the input and the output layer are fixed to be equal to that of input and output variables, whereas the hidden layer can contain more than one layer, and in each layer the number of neurons is flexible. Adjusting the structure of a network, namely the number of hidden layers and neurons, is one of the main ways to improve
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its performance. The structure of network can be expressed as: N in ½N 1 N 2 N h h N out ;
ð1Þ
where Nin and Nout refer to the number of input and output variables, respectively. Subscript h denotes the number of hidden layers, N1, N2 and Nh are the numbers of the neurons in each hidden layer. The network accepts the information from the input layer, processes the data in the hidden layers, and then exports the results via the output layer. In each hidden layer and output layer, the neurons take the output of the neurons in the preceding layer as the input. The data are modulated by transfer function with weights and bias in the neurons to computer the output, as described by: ! X ðnÞ ðn1Þ ðnÞ ðnÞ Xj ¼ f W ji X i þ bj ; ð2Þ i ðnÞ Xj
ðnÞ
where is the output of node j in the nth layer, W ji is the weight from node i in the (n 1)th layer to node j in ðnÞ the nth layer, and bj is the bias of node j in the nth layer. The non-linear transfer function f(x) used in the present work is a tan-sigmoid function: f ðxÞ ¼
1 e2x : 1 þ e2x
Fig. 1. A schematic illustration of an artificial neural network.
ð3Þ
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In order to avoid limiting the output value to a small range, a linear transfer function (f(x) = x) was used in the output layer. Back-propagation (BP) algorithm is an iterative gradient descent approach, which is one of the most widely used training algorithms for multi-layer networks, to minimize the mean squared error E between the predicted and desired values: ð4Þ
where L refers to the number of training patterns, d(t) is the desired output value, and p(t) is the target output value predicted by the ANN for the tth pattern. During the training procedure, the network is presented with the data hundreds of cycles, the weights and biases are adjusted until the expected error level is achieved or the maximum iteration is reached. This iterative adjustment of the weights and biases can be expressed as following: ðnÞ
ðnÞ
M N 1 X 1 X jOp ðiÞ OðiÞj ; M j¼1 N i¼1 OðiÞ
ð6Þ
where M is the repeated times, N is the number of patterns, and Op(i) and O(i) are the ith predicted and measured value, respectively. Obviously, the lower R becomes, the better is the prediction quality of the network. 3.2. Databases and preprocessing
L 1 X 2 E¼ ½dðtÞ pðtÞ ; 2L t¼1
W ji ðkÞ ¼ W ji ðk 1Þ a
R¼
oE ðnÞ
oW ji oE ðnÞ ðnÞ bj ðkÞ ¼ bj ðk 1Þ a ðnÞ ; obj
; ð5Þ
where a represents the learning rate, and k refers to the iteration.
Two databases were used in the present work. One contained 101 independent fretting wear tests of PA 4.6 composites [5]. The other was obtained from RTP Company [6]. It consisted of 93 groups of independent Izod, tension and bending tests of PA 6.6 composites. All measured parameters are listed in Table 1. The input variables include the material compositions (volume or weight fraction of the matrix, the short fibers and the fillers), the testing conditions for PA 4.6 (temperature, normal force, and sliding speed), and the manufacturing process of PA 6.6 composites (impact modification). The output variables include the wear characteristics of PA 4.6 composites (specific wear rate and frictional coefficient), and the mechanical properties of PA 6.6 composites (the Izod impact energy, the tensile and flexural strength, the tensile and flexural modulus). It should be noted here that the following notations were used for the parameter of impact modification: ‘‘1’’, yes; ‘‘0’’, no.
3. Results and discussion 3.1. Application of ANN For practical application, an adequate amount of experimental data is necessary to develop a neural network with good performance. The architecture, learning algorithm and other parameters of the network are optimized aiming at the current problem during training and testing process. Once the network is sufficiently optimized and trained based on these data, it can give the reasonable answers when presented with the input data that it has never experienced. This procedure can be summarized in terms of the following stages [4]: 1. Database collection: collect and pre-process the experimental data to build a large enough database. 2. Optimization of ANN: train the network, test its performance, and then adjust the parameters. 3. Prediction by ANN: use the optimized and trained network to predict new solutions. At the second stage, the experimental database was randomly divided into two parts. One was used to train the network, and the other was used to test it. This training and testing process was repeated independently a certain number of times to avoid the accident error. In order to facilitate the comparisons of performance for different network configurations, a mean relative error was introduced:
Table 1 Measured parameters for input and output of ANN
Input Material compositions
Testing conditions
PA 4.6 composites
PA 6.6 composites
Matrix (69–100 vol.%) Fiber (0–25 vol.%) Filler (0–12 vol.%)
Matrix (40–100 wt.%)
Temperature (20–150 °C) Normal force (10–30 N) Sliding speed (0.02–0.2 m/s)
Manufacturing process
Impact modification (0 or 1)
Output Mechanical properties
Wear characteristics
Fiber (0–60 wt.%) Filler (0–20 wt.%)
Izod impact energy using notched or unnotched specimens Tensile strength Tensile modulus Flexural strength Flexural modulus Specific wear rate Frictional coefficient
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3.3. Optimization of ANN There are many learning algorithms using various strategies to optimize the performance of neural networks, namely minimize error E (Eq. (4)). However, different algorithms perform best on different problems [7]. It is therefore pivotal to choose an appropriate one. According to a preliminary work of the authors [8], five widely-used algorithms provided in the neural network toolbox of MATLAB were studied and compared in this paper. These algorithms are: (a) Powell–Beale conjugate gradient algorithm (CGB), (b) gradient descent algorithm with momentum (GDM), (c) scaled conjugate gradient algorithm (SCG), (d) BFGS quasi-Newton method (BFG) and (e) Levernberg–Marquardt algorithm (LM) (see [9,10] for detailed information). The analysis of performance was done in the same environment and according to two aspects: prediction accuracy and computation cost. For each database mentioned in the previous section, three networks with different hidden layer were used in order to eliminate the disturbance of network structure (Table 2). All these networks have nine input variables (compositions and testing conditions/manufacturing process) and one output variable (specific wear rate for PA 4.6 database and tensile strength for PA 6.6 database). During the evaluation, the networks were trained by 80 data, and tested by 10 data. All data were randomly selected from the databases. For the sake of minimizing the accident error, this training and testing process was repeated 100 times independently. Figs. 2 and 3 give the values of mean relative error R and computation time for each algorithm and network structure, respectively. It is difficult to generalize from these results which algorithm is evidently superior. For example, the GDM algorithm consumed the least time, but its prediction quality was less satisfactory. On the other hand, the SCG algorithm can give the acceptable prediction accuracy at a quite low speed. In the viewpoint of integrating accuracy with efficiency, the CGB algorithm was regarded to fit the present case better than the others. This algorithm gave the highest prediction accuracy among the five algorithms, though it also consumed the most time. However, considering that the training and testing process was actually repeated 100 times, the difference of computation cost of these algorithms for a single training and prediction process was trivial (only several seconds). The CGB algorithm, therefore, will be used in the following work. Defining the architecture of the network is also a very important decision that can dramatically influence the per-
Table 2 Network structures used for analysis of learning algorithm For the database of PA 6.6
For the database of PA 4.6
9-[2]1-1 9-[3]1-1 9-[3-1]2-1
9-[3]1-1 9-[3-1]2-1 9-[9-4-2]3-1
Fig. 2. A comparison of the prediction quality of various learning algorithms: (a) prediction of tensile strength based on PA 6.6 database and (b) prediction of specific wear rate based on PA 4.6 database.
formance of network. However, there is no quantifiable, best answer to construct the hidden layer for a particular application. This makes the task seems more like an ‘‘art’’ for the network designer [11]. Only some general strategies were picked up over time and followed by researchers. It is commonly accepted that increasing the number of neurons can improve the prediction quality of the network. But this number cannot be increased unlimitedly because one may reach a saturation value, resulting in the over-fitting problem. This means the network has just memorized the training examples, but does not learn to generalize to new situations. Though the error on the training data is driven to a very small value, the error will be probably large when new data is presented to the network. In Fig. 4, a network possessing nine input variables (compositions and testing conditions) and one output variable (specific wear rate or frictional coefficient) was used to show the dependence of the prediction accuracy on the network complexity. Here the X-coordinate refers to the network structures, which are listed in Table 3. The evalu-
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Fig. 4. Dependence of mean relative error on the hidden layer architecture for the wear properties of PA 4.6 composites. The X-coordinate represents the network structure no., which can be found in Table 3.
Fig. 3. A comparison of the computation time of various learning algorithms: (a) prediction of tensile strength based on PA 6.6 database and (b) prediction of specific wear rate based on PA 4.6 database.
ation was similar to the above (80 data for training, 10 data for testing, repeated 200 times), nevertheless the prediction accuracy was slightly less than that of the previous analysis
since the maximum number of iterations was reduced by about 30% in order to save computation time. It can be found that, especially for the single hidden layer network, the mean relative error decreases with the increase of neurons in a certain range, and then increases again when more neurons are added. The curves show that a network consisting of a single hidden layer with three neurons gives the best quality for the prediction of frictional coefficient, whereas a structure of 9-[12-6-3]3-1 exhibits a relatively better performance for the specific wear rate of PA 4.6 composites. Although it is proved based on Kolmogorov’s theorem that a two hidden layer network is capable to approximate most functions [12,13], this result probably indicates that to describe the complex relationship between the specific wear rate of SFRPA composites and the chosen input parameters requires a more complicated architecture of the network. Fig. 5 shows the comparison results of PA 6.6 composites through a network with nine inputs (compositions and manufacturing process) and six outputs (strength and modulus). The training and testing procedure
Table 3 Tested ANN architectures No.
Hidden layer structure (for PA 4.6 composites)
Hidden layer structure (for PA 6.6 composites)
No.
Hidden layer structure (for PA 4.6 composites)
Hidden layer structure (for PA 6.6 composites)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
9-[1]1-1 9-[2]1-1 9-[3]1-1 9-[4]1-1 9-[6]1-1 9-[12]1-1 9-[18]1-1 9-[1-1]2-1 9-[2-1]2-1 9-[3-1]2-1 9-[3-3]2-1 9-[4-3]2-1 9-[5-1]2-1 9-[5-3]2-1 9-[6-4]2-1
9-[1]1-6 9-[2]1-6 9-[3]1-6 9-[4]1-6 9-[5]1-6 9-[6]1-6 9-[9]1-6 9-[12]1-6 9-[18]1-6 9-[1-1]2-6 9-[2-1]2-6 9-[3-2]2-6 9-[5-1]2-6 9-[5-4]2-6 9-[8-1]2-6
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
9-[8-2]2-1 9-[10-4]2-1 9-[12-1]2-1 9-[12-4]2-1 9-[13-4]2-1 9-[14-4]2-1 9-[15-2]2-1 9-[4-2-1]3-1 9-[9-4-2]3-1 9-[12-6-3]3-1 9-[12-9-6]3-1 9-[15-9-6]3-1 9-[15-12-6]3-1 9-[30-15-7]3-1 9-[50-25-12]3-1
9-[8-6]2-6 9-[10-1]2-6 9-[10-6]2-6 9-[12-1]2-6 9-[12-6]2-6 9-[15-1]2-6 9-[15-6]2-6 9-[16-8]2-6 9-[4-3-2]3-6 9-[6-4-2]3-6 9-[9-6-3]3-6 9-[12-9-113-6 9-[12-9-6]3-6 9-[15-9-1]3-6 9-[15-12-6]3-6
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Fig. 5. Dependence of mean relative error on the hidden layer architecture for the mechanical properties of PA 6.6 composites. The X-coordinate represents the network structure no., which can be found in Table 3.
Fig. 6. Influence of the output number on the prediction quality for the mechanical properties of PA 6.6 composites ((a) tensile strength and (b) notched Izod impact energy). See Table 3 for the network structure no.
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was as same as the previous one, and the structures can also be found in Table 3. A structure of 9-[5]1-6 was proved to achieve the lowest mean relative error for every output variable. The saturation state was also quite obvious for the single hidden layer network. It should be noticed that the prediction quality of unnotched Izod impact energy is not as good as the other mechanical properties. The difference can be ascribed to the heterogeneity of the intact specimen, which results in a larger fluctuation of the measured results. However, this bias is still in the allowable range comparing with the practical experiments. Multi-dimensional modeling is a highlighted advantage of ANN. Comparing with other phenomenological modeling techniques, e.g. multiple regression analysis, the neural network has a strong capability in exploring the relationship between multiple input and output variables simultaneously. Fig. 6 displays a comparison among the prediction quality of the networks differing by the number
Fig. 7. Influence of the output number on the prediction quality for the wear properties of PA 4.6 composites ((a) specific wear rate and (b) frictional coefficient). See Table 3 for the network structure no.
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Z. Jiang et al. / Materials and Design 29 (2008) 628–637
of output variables for PA 6.6 composites. Although the needed neurons for the highest performance decreased with the reduced number of output variables, the difference of the best results was not conspicuous. Taking account of the computation cost, the multi-output network exhibited a higher efficiency. However, in the case of specific wear rate of PA 4.6, the prediction results of two-output network were definitely inferior to the single-output network, whereas the frictional coefficient seemed to be insensitive to this change, as shown in Fig. 7. Furthermore, it can be found in Figs. 6b and 7a that the mean relative error has a considerable possibility to sharply increase when the number of neuron in the last hidden layer is less than that in the output layer, viz. Nh < Noutput. A one-output neural
Fig. 9. Dependence of the prediction quality on the number of training data for the wear properties of PA 4.6 composites. By the fitting curves (dash lines) a saturation state of prediction quality could be expected when training data number reaches about 180.
Fig. 8. Dependence of prediction quality on the number of training data for the mechanical properties of PA 6.6 composites ((a) notched and unnotched Izod impact energy and (b) tensile strength and modulus, flexural strength and modulus). The fitting curves (dash lines) indicate that a stable mean relative error could be achieved if the training data number reaches about 90.
Fig. 10. Predicted wear characteristics of PA 4.6 composites ((a) specific wear rate and (b) frictional coefficient) as functions of volume percentage of short glass fiber and PTFE filler at 20 °C. Normal force = 10 N, sliding speed = 0.04 m/s. The measured data points are plotted as black dots with error bars.
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network, therefore, is recommended for high prediction quality in practical applications, especially when dealing with some complex problems [14]. Nevertheless a multioutput neural network is more suitable for simpler problems on account of its higher efficiency in computation. A certain amount of training data is an essential condition for a neural network to export prediction results of high accuracy. Figs. 8 and 9 show the dependence of the relative mean error R on the number of training data for mechanical properties of PA 6.6 composites (using the network of 9-[5]1-6), and wear characteristics of PA 4.6 composites (using the network of 9-[12-6-3]3-1 for specific wear rate, and 9-[3]1-1 for frictional coefficient), respectively. The prediction quality increases significantly with the rising number of the training data. By the fitting curves, it can be expected that the relative mean error will be reduced to a stable level when the number of training data reaches approximately 90 (for the mechanical properties of the PA 6.6 composites) or 180 (for the tribological properties of the PA 4.6 composites).
Fig. 11. Predicted 3D profiles show that the dependence of: (a) specific wear rate and (b) frictional coefficient of PA 4.6 composites on testing conditions (normal force and sliding speed) at 20 °C. The measured data points are plotted as black dots with error bars.
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3.4. ANN prediction In the previous sections, the optimal configuration of neural networks has been discussed. For the sake of achieving the highest prediction quality as possible, the optimized networks were trained by the whole experimental database. The new databases used for prediction consisted of only the input variables. Though these input variables had been never experienced by the networks, most of them still fell in the range of the training database because the extrapolation performance of ANN is relatively poor. Fig. 10 shows the predicted 3D profile of the specific wear rate of PA 4.6 composites as a function of the content of short glass fiber and PTFE filler. The measured results are plotted as the dots with error bars. It can be seen that the predicted profile shows a good agreement with the measured points, and furthermore displays a continuous increasing tendency of the wear resistance when the content of glass fiber and PTFE increases. In Fig. 11, the predicted profiles demonstrate the dependence of the wear properties
Fig. 12. Predicted mechanical properties of PA 6.6 composites as functions of volume percentage of short carbon fiber and PTFE filler: (a) notched Izod impact energy, (b) unnotched Izod impact energy, (c) tensile strength, (d) tensile modulus, (e) flexural strength, and (f) flexural modulus. The measured data points are plotted as black dots.
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Fig. 12 (continued)
4. Conclusion
neural networks. With the aid of the predicted 3D profiles, the relationships between the chosen input and output variables have been clearly visualized. The present work shows that the ANN technique can make a more efficient utilization of the relatively limited experimental databases, which means a considerable saving of time and cost for both research and production. It also allows systematic parametric studies on computer for materials optimization since more distinct dependencies of the mechanical and wear characteristics on the compositions and working conditions can be obtained according to the predicted results. The prediction quality has a potential to be improved if the database for training process could be enlarged or the configuration of network could be further optimized. During the development of the network architecture, which has a remarkable influence on the performance of the network, the following practical rules have been summarized:
The ANN technique was applied to predict the mechanical behaviors of PA 6.6 composites and the wear properties of PA 4.6 composites. The properties of these SFRPA composites as a function of the compositions or testing conditions can be well predicted by the optimized and trained
1. In most case, a simple structure of hidden layer including several neurons has an adequate capability to model the problems. However, some complicated relationships, e.g. specific wear rate in this work, require a more complex network structure.
on the testing conditions (sliding speed and normal force). The normal force exhibits a stronger influence on the specific wear rate (Fig. 11a), whereas the frictional coefficient increases more with the rising sliding speed (Fig. 11b). Fig. 12 displays a series of mechanical characteristics of PA 6.6 composites simultaneously predicted by a network. The real data points, marked as black dots, are plotted for contrast. In these 3D profiles, a noticeable enhancement effect of the short carbon fiber can be found. However, the PTFE filler leads to a slight deterioration in strength when its content reaches high values. The prediction quality is quite satisfactory since the dependence of the mechanical behaviors on the compositions is more regular as compared to that of the wear properties.
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2. The determination of best neuron number is case dependent. The prediction quality increases in a certain rage with an increase in neurons, but it deteriorates when the neuron number exceeds the saturation value. 3. Generally speaking, a one-output network can avoid the disturbance of the relationships between inputs and different outputs, and provide a higher prediction quality. Nevertheless, a multi-output network is still recommended for some relatively simple problems, such as prediction of mechanical properties, because the difference of the quality between the one-output and multioutput network is negligible, and the obviously less computation time is needed by the latter.
Acknowledgements The authors acknowledge the support of DFG (FR 675/ 45-1). Z. Zhang is grateful to the Alexander von Humboldt Foundation for his Sofja Kovalevskaja Award, financed by the German Federal Ministry of Education and Research within the German Government’s investment in the future program. References [1] Zeng P. Neural computing in mechanics. Appl Mech Rev 1998;51: 173–97.
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[2] Zhang Z, Friedrich K. Artificial neural networks applied to polymer composites: a review. Compos Sci Technol 2003;63:2029–44. [3] Kadi HE. Modeling the mechanical behavior of fiber-reinforced polymeric composite materials using artificial neural networks—a review. Compos Struct 2006;73:1–23. [4] Lee JA, Almond DP, Harris B. The use of neural networks for the prediction of fatigue lives of composite materials. Composites A 1999;30:1159–69. [5] Reinicke R, Haupert F, Friedrich K. On the tribological behavior of selected, injection molded thermoplastics. Composites A 1998;29: 763–71. [6] http://www.rtpcompany.com/,RTP Company. [7] Koker R, Altinkok N, Demir A. Neural network based prediction of mechanical properties of particulate reinforced metal matrix composites using various training algorithms. Mater Design 2007;28: 616–27. [8] Jiang Z, Zhang Z, Friedrich K. Prediction on wear properties of polymer composites with artificial neural networks. Compos Sci Technol 2007;67:168–76. [9] Demuth H, Beale M. Neural network toolbox: for use with MATLAB (Version 4.0). The MathWorks, Inc.; 2004. [10] Haykin S. Neural networks: a comprehensive foundation. Upper Saddle River: Prentice Hall; 1999. [11] Anderson D, McNeill G. Artificial neural networks technology. Kaman Sciences Corporation; 1992. [12] Kurkova´ V. Kolmogrov’s theorem is relevant. Neural Comput 1991;3:617–22. [13] Swingler K. Applying neural networks: a practical guide. San Francisco: Morgan Kaufman Publishers, Inc.; 1996. [14] Zhang Z, Klein P, Friedrich K. Dynamic mechanical properties of PTFE based short carbon fibre reinforced composites: experiment and artificial neural network prediction. Compos Sci Technol 2002;62:1001–9.
Materials & Design
Materials and Design 29 (2008) 628–637 www.elsevier.com/locate/matdes
Neural network based prediction on mechanical and wear properties of short fibers reinforced polyamide composites Zhenyu Jiang a, Lada Gyurova a, Zhong Zhang a
b,*
, Klaus Friedrich a, Alois K. Schlarb
a
Institute for Composite Materials (Institut fuer Verbundwerkstoffe), University of Kaiserslautern, 67663 Kaiserslautern, Germany b National Center for Nanoscience and Technology, No. 2, 1st North Street, Zhongguancun, 100080 Beijing, China Received 11 September 2006; accepted 14 February 2007 Available online 2 March 2007
Abstract The artificial neural network technique was applied to predict the mechanical and wear properties of short fiber reinforced polyamide (PA) composites. Two experimental databases were used to train the neural network: one consisted of 101 independent fretting wear tests of PA 4.6 composites; the other one was from a commercial company and included 93 pairs of independent Izod impact, tension and bending tests of PA 6.6 composites. The predicted property profiles as a function of short fiber content or testing conditions proved a remarkable capability of well-optimized neural networks for modeling concern. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Polymer matrix; Mechanical and wear properties; Short fiber; Artificial neural network
1. Introduction Polyamide (PA) is one kind of the most versatile engineering thermoplastics, for their outstanding performance and ease of processing. It is well known that with the reinforcement of short fibers the mechanical strength and wear resistance of the polyamide composites can be significantly improved. This makes short fiber reinforced polyamide composites (SFRPA) a superior candidate for replacement of metal materials in various cases. Therefore, the design of SFRPA to meet the desired target is of importance for the present engineering application. However, the mechanical behavior and wear characteristics of SFRPA composites are currently regarded as a complicated problem, which is associated with multifold factors, e.g. composition, working condition, and manufacturing process. The property investigation of SFRPA composites is difficult to analyze theoretically, and remains to be a labor intensive and time-consuming problem for experimental research.
*
Corresponding author. Tel.: +86 10 62652699; fax: +86 10 62650450. E-mail address: [email protected] (Z. Zhang).
0261-3069/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2007.02.008
In recent years, the application of artificial neural network (ANN) has attracted extensive interests in diverse fields. Inspired by its biological counterpart, ANN uses the inter-connected mathematical nodes to imitate some abilities of human brain. Such a network can learn (or be trained) from examples, and explore by itself the underlying functional relationships between the given reasons and results. This remarkable capability of modeling is useful in the study of complicated problems, which usually cannot be solved by existing physical theories or the other mathematical approaches. Especially, the ability to find the correlation among multiple parameters with noise and to implement excellent non-linear interpolation are extremely required by industrial community for the estimation of performance of the materials with new combination of composition or processing based on the existing experimental databases. Therefore, the ANN technique has so far taken a great deal of attention in the research on mechanical and wear behaviors of polymer composites (cf. [1–3] for reviews). In the present work, the ANN with back-propagation (BP) algorithm is applied to predict the mechanical or wear properties of two kinds of SFRPA composites respectively:
Z. Jiang et al. / Materials and Design 29 (2008) 628–637
PA 6.6 and PA 4.6 composites, both of which are reinforced by short glass or carbon fibers. Based on these databases, the optimization of the neural networks is discussed through a series of comparisons. Some practical rules have been summarized. Finally the well-optimized and trained neural networks are used to predict the profile of mechanical or wear properties as a function of the content of short fiber/filler or testing condition, according to the newly constructed input data sets. 2. Neural network approach An ANN structure is commonly divided into three parts: input layer, hidden layer and output layer, as illustrated in Fig. 1. The nodes (the basic processing units, also called neurons) are connected by weighted inter-connections, which resemble the intensity of the bioelectricity transferring among the neuron cells in a real neural network. The learned knowledge can be memorized in terms of the state of these weights as well as the biases. The numbers of neurons in the input and the output layer are fixed to be equal to that of input and output variables, whereas the hidden layer can contain more than one layer, and in each layer the number of neurons is flexible. Adjusting the structure of a network, namely the number of hidden layers and neurons, is one of the main ways to improve
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its performance. The structure of network can be expressed as: N in ½N 1 N 2 N h h N out ;
ð1Þ
where Nin and Nout refer to the number of input and output variables, respectively. Subscript h denotes the number of hidden layers, N1, N2 and Nh are the numbers of the neurons in each hidden layer. The network accepts the information from the input layer, processes the data in the hidden layers, and then exports the results via the output layer. In each hidden layer and output layer, the neurons take the output of the neurons in the preceding layer as the input. The data are modulated by transfer function with weights and bias in the neurons to computer the output, as described by: ! X ðnÞ ðn1Þ ðnÞ ðnÞ Xj ¼ f W ji X i þ bj ; ð2Þ i ðnÞ Xj
ðnÞ
where is the output of node j in the nth layer, W ji is the weight from node i in the (n 1)th layer to node j in ðnÞ the nth layer, and bj is the bias of node j in the nth layer. The non-linear transfer function f(x) used in the present work is a tan-sigmoid function: f ðxÞ ¼
1 e2x : 1 þ e2x
Fig. 1. A schematic illustration of an artificial neural network.
ð3Þ
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In order to avoid limiting the output value to a small range, a linear transfer function (f(x) = x) was used in the output layer. Back-propagation (BP) algorithm is an iterative gradient descent approach, which is one of the most widely used training algorithms for multi-layer networks, to minimize the mean squared error E between the predicted and desired values: ð4Þ
where L refers to the number of training patterns, d(t) is the desired output value, and p(t) is the target output value predicted by the ANN for the tth pattern. During the training procedure, the network is presented with the data hundreds of cycles, the weights and biases are adjusted until the expected error level is achieved or the maximum iteration is reached. This iterative adjustment of the weights and biases can be expressed as following: ðnÞ
ðnÞ
M N 1 X 1 X jOp ðiÞ OðiÞj ; M j¼1 N i¼1 OðiÞ
ð6Þ
where M is the repeated times, N is the number of patterns, and Op(i) and O(i) are the ith predicted and measured value, respectively. Obviously, the lower R becomes, the better is the prediction quality of the network. 3.2. Databases and preprocessing
L 1 X 2 E¼ ½dðtÞ pðtÞ ; 2L t¼1
W ji ðkÞ ¼ W ji ðk 1Þ a
R¼
oE ðnÞ
oW ji oE ðnÞ ðnÞ bj ðkÞ ¼ bj ðk 1Þ a ðnÞ ; obj
; ð5Þ
where a represents the learning rate, and k refers to the iteration.
Two databases were used in the present work. One contained 101 independent fretting wear tests of PA 4.6 composites [5]. The other was obtained from RTP Company [6]. It consisted of 93 groups of independent Izod, tension and bending tests of PA 6.6 composites. All measured parameters are listed in Table 1. The input variables include the material compositions (volume or weight fraction of the matrix, the short fibers and the fillers), the testing conditions for PA 4.6 (temperature, normal force, and sliding speed), and the manufacturing process of PA 6.6 composites (impact modification). The output variables include the wear characteristics of PA 4.6 composites (specific wear rate and frictional coefficient), and the mechanical properties of PA 6.6 composites (the Izod impact energy, the tensile and flexural strength, the tensile and flexural modulus). It should be noted here that the following notations were used for the parameter of impact modification: ‘‘1’’, yes; ‘‘0’’, no.
3. Results and discussion 3.1. Application of ANN For practical application, an adequate amount of experimental data is necessary to develop a neural network with good performance. The architecture, learning algorithm and other parameters of the network are optimized aiming at the current problem during training and testing process. Once the network is sufficiently optimized and trained based on these data, it can give the reasonable answers when presented with the input data that it has never experienced. This procedure can be summarized in terms of the following stages [4]: 1. Database collection: collect and pre-process the experimental data to build a large enough database. 2. Optimization of ANN: train the network, test its performance, and then adjust the parameters. 3. Prediction by ANN: use the optimized and trained network to predict new solutions. At the second stage, the experimental database was randomly divided into two parts. One was used to train the network, and the other was used to test it. This training and testing process was repeated independently a certain number of times to avoid the accident error. In order to facilitate the comparisons of performance for different network configurations, a mean relative error was introduced:
Table 1 Measured parameters for input and output of ANN
Input Material compositions
Testing conditions
PA 4.6 composites
PA 6.6 composites
Matrix (69–100 vol.%) Fiber (0–25 vol.%) Filler (0–12 vol.%)
Matrix (40–100 wt.%)
Temperature (20–150 °C) Normal force (10–30 N) Sliding speed (0.02–0.2 m/s)
Manufacturing process
Impact modification (0 or 1)
Output Mechanical properties
Wear characteristics
Fiber (0–60 wt.%) Filler (0–20 wt.%)
Izod impact energy using notched or unnotched specimens Tensile strength Tensile modulus Flexural strength Flexural modulus Specific wear rate Frictional coefficient
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3.3. Optimization of ANN There are many learning algorithms using various strategies to optimize the performance of neural networks, namely minimize error E (Eq. (4)). However, different algorithms perform best on different problems [7]. It is therefore pivotal to choose an appropriate one. According to a preliminary work of the authors [8], five widely-used algorithms provided in the neural network toolbox of MATLAB were studied and compared in this paper. These algorithms are: (a) Powell–Beale conjugate gradient algorithm (CGB), (b) gradient descent algorithm with momentum (GDM), (c) scaled conjugate gradient algorithm (SCG), (d) BFGS quasi-Newton method (BFG) and (e) Levernberg–Marquardt algorithm (LM) (see [9,10] for detailed information). The analysis of performance was done in the same environment and according to two aspects: prediction accuracy and computation cost. For each database mentioned in the previous section, three networks with different hidden layer were used in order to eliminate the disturbance of network structure (Table 2). All these networks have nine input variables (compositions and testing conditions/manufacturing process) and one output variable (specific wear rate for PA 4.6 database and tensile strength for PA 6.6 database). During the evaluation, the networks were trained by 80 data, and tested by 10 data. All data were randomly selected from the databases. For the sake of minimizing the accident error, this training and testing process was repeated 100 times independently. Figs. 2 and 3 give the values of mean relative error R and computation time for each algorithm and network structure, respectively. It is difficult to generalize from these results which algorithm is evidently superior. For example, the GDM algorithm consumed the least time, but its prediction quality was less satisfactory. On the other hand, the SCG algorithm can give the acceptable prediction accuracy at a quite low speed. In the viewpoint of integrating accuracy with efficiency, the CGB algorithm was regarded to fit the present case better than the others. This algorithm gave the highest prediction accuracy among the five algorithms, though it also consumed the most time. However, considering that the training and testing process was actually repeated 100 times, the difference of computation cost of these algorithms for a single training and prediction process was trivial (only several seconds). The CGB algorithm, therefore, will be used in the following work. Defining the architecture of the network is also a very important decision that can dramatically influence the per-
Table 2 Network structures used for analysis of learning algorithm For the database of PA 6.6
For the database of PA 4.6
9-[2]1-1 9-[3]1-1 9-[3-1]2-1
9-[3]1-1 9-[3-1]2-1 9-[9-4-2]3-1
Fig. 2. A comparison of the prediction quality of various learning algorithms: (a) prediction of tensile strength based on PA 6.6 database and (b) prediction of specific wear rate based on PA 4.6 database.
formance of network. However, there is no quantifiable, best answer to construct the hidden layer for a particular application. This makes the task seems more like an ‘‘art’’ for the network designer [11]. Only some general strategies were picked up over time and followed by researchers. It is commonly accepted that increasing the number of neurons can improve the prediction quality of the network. But this number cannot be increased unlimitedly because one may reach a saturation value, resulting in the over-fitting problem. This means the network has just memorized the training examples, but does not learn to generalize to new situations. Though the error on the training data is driven to a very small value, the error will be probably large when new data is presented to the network. In Fig. 4, a network possessing nine input variables (compositions and testing conditions) and one output variable (specific wear rate or frictional coefficient) was used to show the dependence of the prediction accuracy on the network complexity. Here the X-coordinate refers to the network structures, which are listed in Table 3. The evalu-
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Fig. 4. Dependence of mean relative error on the hidden layer architecture for the wear properties of PA 4.6 composites. The X-coordinate represents the network structure no., which can be found in Table 3.
Fig. 3. A comparison of the computation time of various learning algorithms: (a) prediction of tensile strength based on PA 6.6 database and (b) prediction of specific wear rate based on PA 4.6 database.
ation was similar to the above (80 data for training, 10 data for testing, repeated 200 times), nevertheless the prediction accuracy was slightly less than that of the previous analysis
since the maximum number of iterations was reduced by about 30% in order to save computation time. It can be found that, especially for the single hidden layer network, the mean relative error decreases with the increase of neurons in a certain range, and then increases again when more neurons are added. The curves show that a network consisting of a single hidden layer with three neurons gives the best quality for the prediction of frictional coefficient, whereas a structure of 9-[12-6-3]3-1 exhibits a relatively better performance for the specific wear rate of PA 4.6 composites. Although it is proved based on Kolmogorov’s theorem that a two hidden layer network is capable to approximate most functions [12,13], this result probably indicates that to describe the complex relationship between the specific wear rate of SFRPA composites and the chosen input parameters requires a more complicated architecture of the network. Fig. 5 shows the comparison results of PA 6.6 composites through a network with nine inputs (compositions and manufacturing process) and six outputs (strength and modulus). The training and testing procedure
Table 3 Tested ANN architectures No.
Hidden layer structure (for PA 4.6 composites)
Hidden layer structure (for PA 6.6 composites)
No.
Hidden layer structure (for PA 4.6 composites)
Hidden layer structure (for PA 6.6 composites)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
9-[1]1-1 9-[2]1-1 9-[3]1-1 9-[4]1-1 9-[6]1-1 9-[12]1-1 9-[18]1-1 9-[1-1]2-1 9-[2-1]2-1 9-[3-1]2-1 9-[3-3]2-1 9-[4-3]2-1 9-[5-1]2-1 9-[5-3]2-1 9-[6-4]2-1
9-[1]1-6 9-[2]1-6 9-[3]1-6 9-[4]1-6 9-[5]1-6 9-[6]1-6 9-[9]1-6 9-[12]1-6 9-[18]1-6 9-[1-1]2-6 9-[2-1]2-6 9-[3-2]2-6 9-[5-1]2-6 9-[5-4]2-6 9-[8-1]2-6
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
9-[8-2]2-1 9-[10-4]2-1 9-[12-1]2-1 9-[12-4]2-1 9-[13-4]2-1 9-[14-4]2-1 9-[15-2]2-1 9-[4-2-1]3-1 9-[9-4-2]3-1 9-[12-6-3]3-1 9-[12-9-6]3-1 9-[15-9-6]3-1 9-[15-12-6]3-1 9-[30-15-7]3-1 9-[50-25-12]3-1
9-[8-6]2-6 9-[10-1]2-6 9-[10-6]2-6 9-[12-1]2-6 9-[12-6]2-6 9-[15-1]2-6 9-[15-6]2-6 9-[16-8]2-6 9-[4-3-2]3-6 9-[6-4-2]3-6 9-[9-6-3]3-6 9-[12-9-113-6 9-[12-9-6]3-6 9-[15-9-1]3-6 9-[15-12-6]3-6
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Fig. 5. Dependence of mean relative error on the hidden layer architecture for the mechanical properties of PA 6.6 composites. The X-coordinate represents the network structure no., which can be found in Table 3.
Fig. 6. Influence of the output number on the prediction quality for the mechanical properties of PA 6.6 composites ((a) tensile strength and (b) notched Izod impact energy). See Table 3 for the network structure no.
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was as same as the previous one, and the structures can also be found in Table 3. A structure of 9-[5]1-6 was proved to achieve the lowest mean relative error for every output variable. The saturation state was also quite obvious for the single hidden layer network. It should be noticed that the prediction quality of unnotched Izod impact energy is not as good as the other mechanical properties. The difference can be ascribed to the heterogeneity of the intact specimen, which results in a larger fluctuation of the measured results. However, this bias is still in the allowable range comparing with the practical experiments. Multi-dimensional modeling is a highlighted advantage of ANN. Comparing with other phenomenological modeling techniques, e.g. multiple regression analysis, the neural network has a strong capability in exploring the relationship between multiple input and output variables simultaneously. Fig. 6 displays a comparison among the prediction quality of the networks differing by the number
Fig. 7. Influence of the output number on the prediction quality for the wear properties of PA 4.6 composites ((a) specific wear rate and (b) frictional coefficient). See Table 3 for the network structure no.
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of output variables for PA 6.6 composites. Although the needed neurons for the highest performance decreased with the reduced number of output variables, the difference of the best results was not conspicuous. Taking account of the computation cost, the multi-output network exhibited a higher efficiency. However, in the case of specific wear rate of PA 4.6, the prediction results of two-output network were definitely inferior to the single-output network, whereas the frictional coefficient seemed to be insensitive to this change, as shown in Fig. 7. Furthermore, it can be found in Figs. 6b and 7a that the mean relative error has a considerable possibility to sharply increase when the number of neuron in the last hidden layer is less than that in the output layer, viz. Nh < Noutput. A one-output neural
Fig. 9. Dependence of the prediction quality on the number of training data for the wear properties of PA 4.6 composites. By the fitting curves (dash lines) a saturation state of prediction quality could be expected when training data number reaches about 180.
Fig. 8. Dependence of prediction quality on the number of training data for the mechanical properties of PA 6.6 composites ((a) notched and unnotched Izod impact energy and (b) tensile strength and modulus, flexural strength and modulus). The fitting curves (dash lines) indicate that a stable mean relative error could be achieved if the training data number reaches about 90.
Fig. 10. Predicted wear characteristics of PA 4.6 composites ((a) specific wear rate and (b) frictional coefficient) as functions of volume percentage of short glass fiber and PTFE filler at 20 °C. Normal force = 10 N, sliding speed = 0.04 m/s. The measured data points are plotted as black dots with error bars.
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network, therefore, is recommended for high prediction quality in practical applications, especially when dealing with some complex problems [14]. Nevertheless a multioutput neural network is more suitable for simpler problems on account of its higher efficiency in computation. A certain amount of training data is an essential condition for a neural network to export prediction results of high accuracy. Figs. 8 and 9 show the dependence of the relative mean error R on the number of training data for mechanical properties of PA 6.6 composites (using the network of 9-[5]1-6), and wear characteristics of PA 4.6 composites (using the network of 9-[12-6-3]3-1 for specific wear rate, and 9-[3]1-1 for frictional coefficient), respectively. The prediction quality increases significantly with the rising number of the training data. By the fitting curves, it can be expected that the relative mean error will be reduced to a stable level when the number of training data reaches approximately 90 (for the mechanical properties of the PA 6.6 composites) or 180 (for the tribological properties of the PA 4.6 composites).
Fig. 11. Predicted 3D profiles show that the dependence of: (a) specific wear rate and (b) frictional coefficient of PA 4.6 composites on testing conditions (normal force and sliding speed) at 20 °C. The measured data points are plotted as black dots with error bars.
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3.4. ANN prediction In the previous sections, the optimal configuration of neural networks has been discussed. For the sake of achieving the highest prediction quality as possible, the optimized networks were trained by the whole experimental database. The new databases used for prediction consisted of only the input variables. Though these input variables had been never experienced by the networks, most of them still fell in the range of the training database because the extrapolation performance of ANN is relatively poor. Fig. 10 shows the predicted 3D profile of the specific wear rate of PA 4.6 composites as a function of the content of short glass fiber and PTFE filler. The measured results are plotted as the dots with error bars. It can be seen that the predicted profile shows a good agreement with the measured points, and furthermore displays a continuous increasing tendency of the wear resistance when the content of glass fiber and PTFE increases. In Fig. 11, the predicted profiles demonstrate the dependence of the wear properties
Fig. 12. Predicted mechanical properties of PA 6.6 composites as functions of volume percentage of short carbon fiber and PTFE filler: (a) notched Izod impact energy, (b) unnotched Izod impact energy, (c) tensile strength, (d) tensile modulus, (e) flexural strength, and (f) flexural modulus. The measured data points are plotted as black dots.
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Fig. 12 (continued)
4. Conclusion
neural networks. With the aid of the predicted 3D profiles, the relationships between the chosen input and output variables have been clearly visualized. The present work shows that the ANN technique can make a more efficient utilization of the relatively limited experimental databases, which means a considerable saving of time and cost for both research and production. It also allows systematic parametric studies on computer for materials optimization since more distinct dependencies of the mechanical and wear characteristics on the compositions and working conditions can be obtained according to the predicted results. The prediction quality has a potential to be improved if the database for training process could be enlarged or the configuration of network could be further optimized. During the development of the network architecture, which has a remarkable influence on the performance of the network, the following practical rules have been summarized:
The ANN technique was applied to predict the mechanical behaviors of PA 6.6 composites and the wear properties of PA 4.6 composites. The properties of these SFRPA composites as a function of the compositions or testing conditions can be well predicted by the optimized and trained
1. In most case, a simple structure of hidden layer including several neurons has an adequate capability to model the problems. However, some complicated relationships, e.g. specific wear rate in this work, require a more complex network structure.
on the testing conditions (sliding speed and normal force). The normal force exhibits a stronger influence on the specific wear rate (Fig. 11a), whereas the frictional coefficient increases more with the rising sliding speed (Fig. 11b). Fig. 12 displays a series of mechanical characteristics of PA 6.6 composites simultaneously predicted by a network. The real data points, marked as black dots, are plotted for contrast. In these 3D profiles, a noticeable enhancement effect of the short carbon fiber can be found. However, the PTFE filler leads to a slight deterioration in strength when its content reaches high values. The prediction quality is quite satisfactory since the dependence of the mechanical behaviors on the compositions is more regular as compared to that of the wear properties.
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2. The determination of best neuron number is case dependent. The prediction quality increases in a certain rage with an increase in neurons, but it deteriorates when the neuron number exceeds the saturation value. 3. Generally speaking, a one-output network can avoid the disturbance of the relationships between inputs and different outputs, and provide a higher prediction quality. Nevertheless, a multi-output network is still recommended for some relatively simple problems, such as prediction of mechanical properties, because the difference of the quality between the one-output and multioutput network is negligible, and the obviously less computation time is needed by the latter.
Acknowledgements The authors acknowledge the support of DFG (FR 675/ 45-1). Z. Zhang is grateful to the Alexander von Humboldt Foundation for his Sofja Kovalevskaja Award, financed by the German Federal Ministry of Education and Research within the German Government’s investment in the future program. References [1] Zeng P. Neural computing in mechanics. Appl Mech Rev 1998;51: 173–97.
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[2] Zhang Z, Friedrich K. Artificial neural networks applied to polymer composites: a review. Compos Sci Technol 2003;63:2029–44. [3] Kadi HE. Modeling the mechanical behavior of fiber-reinforced polymeric composite materials using artificial neural networks—a review. Compos Struct 2006;73:1–23. [4] Lee JA, Almond DP, Harris B. The use of neural networks for the prediction of fatigue lives of composite materials. Composites A 1999;30:1159–69. [5] Reinicke R, Haupert F, Friedrich K. On the tribological behavior of selected, injection molded thermoplastics. Composites A 1998;29: 763–71. [6] http://www.rtpcompany.com/,RTP Company. [7] Koker R, Altinkok N, Demir A. Neural network based prediction of mechanical properties of particulate reinforced metal matrix composites using various training algorithms. Mater Design 2007;28: 616–27. [8] Jiang Z, Zhang Z, Friedrich K. Prediction on wear properties of polymer composites with artificial neural networks. Compos Sci Technol 2007;67:168–76. [9] Demuth H, Beale M. Neural network toolbox: for use with MATLAB (Version 4.0). The MathWorks, Inc.; 2004. [10] Haykin S. Neural networks: a comprehensive foundation. Upper Saddle River: Prentice Hall; 1999. [11] Anderson D, McNeill G. Artificial neural networks technology. Kaman Sciences Corporation; 1992. [12] Kurkova´ V. Kolmogrov’s theorem is relevant. Neural Comput 1991;3:617–22. [13] Swingler K. Applying neural networks: a practical guide. San Francisco: Morgan Kaufman Publishers, Inc.; 1996. [14] Zhang Z, Klein P, Friedrich K. Dynamic mechanical properties of PTFE based short carbon fibre reinforced composites: experiment and artificial neural network prediction. Compos Sci Technol 2002;62:1001–9.