ORIGINAL RESEARCH

Modelling the correlation between the geometrical featuresand the forming limit strains of perforated Al 8011 sheetsusing artificial neural network

K. Elangovan & C. Sathiya Narayanan &

R. Narayanasamy

Received: 1 June 2010 /Accepted: 8 October 2010 /Published online: 16 November 2010# Springer-Verlag France 2010

Abstract In the recent years, sheet metals are produced withperforations in various shapes and patterns to improve theappearance of sheet and to save weight of components. As inconventional metal sheets, it is important to form theperforated sheet metals also within their safe strain regionsto avoid the forming failures like necking, fracture andwrinkling. The Forming Limit Diagram (FLD) is an appro-priate tool to determine the forming limit strains. The limitingstrains of perforated sheet metals mainly depend on thegeometry of the perforations and forming variables. This leadsto large increase in number of test to be conducted withvarious geometry and forming variables for determining theforming limit strain for perforated sheets. Aiming to reducethe number of experiments needed, in this work, an ArtificialNeural Network (ANN) model has been developed forforming limit diagram of perforated Al 8011 sheets based onexperimental results and correlated with the geometricalfeatures of the perforated sheets. This model is a feed forwardback propagation neural network (BPNN) with a set ofgeometrical variables as its inputs and the safe true strains asits output. This ANN model can be applied for prediction ofFLD of perforated sheet having any geometry.

Keywords Forming limit diagram . Artificial neuralnetwork . Perforated Al 8011

Introduction

Forming limit diagram (FLD), an effective tool forevaluating the formability of a sheet metal for variousstrain states and various strain ratios, can be used forfinding out the limiting major strain and limiting minorstrain for any strain states and strain ratios. These forminglimit diagrams are being used by both manufacturers andusers of the sheet metals. From the initiation of FLDconcept, many numbers of works have been carried out inthis field of FLD of sheet metals [1–3]. The importantlimitation of the FLD is that a FLD is applicable only for aparticular sheet quality [4]. As the chemical composition,processing method, thickness, heat treatment after manu-facture of sheet metal, strain path, strain condition and etc.,are varied, there will be variation in limiting strain for the sheetwhich alters the FLD suitably. The construction of a FLD thatsuits for all above variation needs a lot of experimental work tobe carried out. With the aim to reduce the experimental workfor constructing FLD, many researchers have developed no ofmathematical models [5–7]. In recent years, researchers areattempting to develop ANN model, which has universalapproximation capability and more accurate in predicting anysystem [8]. This ANN is also being used for sheet metalproblems. The application of ANN for sheet metal problemsincluding the following works.

Sivasankaran et al. [9] developed an ANN model ofdeep drawing process which predicts the onset ofwrinkling for the commercial pure aluminium sheetsdrawn through conical die by considering mechanicalproperties, annealing temperature, soaking time, blankdiameter, and number of drawn as input neurons for ANNmodel. Wang et al. [10] generalized and correlated thegeometrical features of various formed parts are withsurface behaviour and a model based on an artificial neural

K. Elangovan : C. S. Narayanan (*) :R. NarayanasamyDepartment of Production Engineering,National Institute of Technology,Trichy,620 015 Tamilnadu, Indiae-mail: csathiya@nitt.edu

K. Elangovane-mail: elangs77@yahoo.co.in

R. Narayanasamye-mail: narayan@nitt.edu

Int J Mater Form (2011) 4:389–399DOI 10.1007/s12289-010-1003-x

network is introduced to reveal the critical geometricalcriteria for wrinkling. Veera Babu et al. [11] developed anartificial neural network model to predict the tensilebehavior of tailor welded blanks made of steel grade andaluminium alloy base materials. Zemin Fu et al. [12] usedneural network to predict punch radius based on the resultsof air-bending experiments of sheet metal which was thethree-layer back propagation neural network. They used agenetic algorithm (GA) to optimize the weights of neuralnetwork for minimizing the error between the predictivepunch radius and the experimental one. Wenjuan Liu et al.[13] proposed a technique based on artificial neuralnetwork and genetic algorithm to solve the problem ofspringback and used an improved genetic algorithm tooptimize the weights of neural network. In addition to thiswork Recep Kazan et al. [14] developed the model ofspringback in wipe-bending process using artificial neuralnetwork approach. Swadesh Kumar Singh et al. [15] andCapdevila et al. [16] developed a neural network modelfor describing the evolution of mechanical properties suchas yield strength, ultimate tensile strength and elongationon low carbon sheet steels.

However, the application of ANN is very limited inpredicting FLD for sheet metals. In sheet metal field,perforated sheets are being produced in order to reduce theweight of the components and to import aesthetic appear-ance and components are being made out of perforatedsheets by forming operations. Such perforations havenegative effects on FLD of sheet metals. In spite of thesedisadvantages, perforated sheets are being used for theabove said advantages. The perforated aluminium sheetsare widely used in automotive, acoustics, architecture,interior design, displays, manufacturing, safety flooring,agriculture, pollution control, transportation, and miningetc. [17]. Therefore the manufacturers/users are in aposition to develop FLD for the perforated sheet metalsalso. Perforations are made in various patterns and variousdimensions.

The deformation behaviour of perforated sheets duringmetal forming operations is different from that of plain sheetmetals and this had been studied by chen and coworkers [18,19] and baik et al. [20–22]. In attempt to study theformability of perforated sheets, Yi-Che Lee et al. [23]developed yield criterion for perforated sheets in terms ofapparent stresses by employing the equivalent–continuationapproach. However it is focuses on perforated sheets withlow ligament ratios. The above works on perforated sheetswere limited to the perforated sheets having one or two typesof geometrical features. Therefore it is needed to develop amodel which accommodates all the variations. Among thealuminium alloys of various grades, Al 8000 series whichhas more strength is preferred by all the manufacturers/userseven though the strength of the perforated sheets are less.

This makes the authors to develop a model for the predictionof FLD of perforated Al 8011 sheets. In this work, it isconcentrated on the development of ANN model that suits topredict the formability of Al 8011 alloy sheets with variouscombinations of thicknesses, size of circular perforations andligament widths.

Experimental data collection

Chemical composition

The chemical composition of the aluminium alloy 8011 sheettaken for this study is given in Table 1. The aluminium alloy8011 grade contains a smaller percentage of iron (Fe) andsilicon (Si), which increases the strength of aluminium [24].In addition to this, it contains very little amount ofmanganese, which increases the corrosion resistance.

Specifications of perforated sheet blanks

Perforated sheet blanks were prepared from solid Al 8011sheet with various combination of diameter of circularperforation and ligament width by punching operation inthe straight pattern as shown in Fig. 1. The differentspecifications of the 27 set of perforated Al 8011 blankstaken for forming limit diagram study are shown in Table 2.

Forming limit diagram

As in a standard FLD test [3], blanks of dimensions 300×200, 300×180, 300×160, 300×140, 300×120, 300×120,300×100 and 300×80mm were sheared from the preparedperforated sheets by shearing. Samples were formed up tofracture using a double action hydraulic press of capacity2,000 kN with standard die and punch set up. A typical

Table 1 Chemical composition of Al 8011 (%W)

Al Fe Si Mn Zn Cr Mg

Reminder 0.797 0.433 0.131 0.033 0.011 0.109

Fig. 1 Straight pattern ofperforations

390 Int J Mater Form (2011) 4:389–399

punch-die assembly used in the experiments is shown inFig. 2. As the perforated sheet samples were subjected todifferent conditions of strain namely tension–tension, planestrain and tension–compression by varying width of thesamples during forming. During forming the perforatedsheets deformed plastically. Before and after forming, thedistances between two equivalent points along longitudinal

(major) and transverse (minor) directions were measured inthe necked region [25]. The major true strain (ε1) and minortrue strain (ε2) are then calculated using the formulae givenin Eqs. 1 and 2 respectively. FLD was drawn by plotting theminor strain in abscissa and corresponding major strain inordinate and by drawing a smooth curve connecting thesepoints [26].

Major true strain ð"1Þ ¼ lnDistance between two equivalent points after forming in the major axis

Distance between two equivalent points before forming in the major axis

� �ð1Þ

Minor true strain ð"2Þ ¼ lnDistance between two equivalent points after forming in the minor axis

Distance between two equivalent points before forming in the minor axis

� �ð2Þ

Artificial neural network

The artificial neural networks (ANN) have been developed bygeneralization of the mathematical model of the human brain’sability and neural biology. An ANN is a set of processingelements or neurons, and connections with adjustable weights.One of the advantages of using the neural network approach isthat a model can be constructed very easily based on the giveninput and output and trained to accurately predict processdynamics [5, 6]. This modeling technique is particularlyprecious in processes where a complete understanding of thephysical mechanisms is very complicated as in the case ofsheet metal forming process. The neural network learns byusing training data. Input variables are supplied and theresultant output is compared with the desired output. Thenetwork then adjusts the interconnection weights betweenlayers. This process is repeated until the network performswell on the training set. The network can then be assessed ondata not included in the training set, the validation data, toestimate its performance in order to achieve better networkconvergence.

Backpropagation (BP) network

There exist many different architecture and learningalgorithms for neural network models. Most successfuland powerful network is three layer backpropagationnetwork designs [27], and is described with the help offlow diagram as shown in Fig. 3. Neural network is alogical structure with multi-processing elements, which areconnected through interconnection weights. The knowledgeis presented by the interconnection weights, which are

Table 2 Specifications of perforated Al 8011 sheets

Sl no Thickness(mm)

Diameter ofperforation (mm)

Ligamentwidth (mm)

1 2.0 3.0 5.0

2 2.0 3.0 7.5

3 2.0 3.0 10.0

4 2.0 4.0 5.0

5 2.0 4.0 7.5

6 2.0 4.0 10.0

7 2.0 5.0 5.0

8 2.0 5.0 7.5

9 2.0 5.0 10.0

10 1.6 3.0 5.0

11 1.6 3.0 7.5

12 1.6 3.0 10.0

13 1.6 4.0 5.0

14 1.6 4.0 7.5

15 1.6 4.0 10.0

16 1.6 5.0 5.0

17 1.6 5.0 7.5

18 1.6 5.0 10.0

19 1.0 3.0 5.0

20 1.0 3.0 7.5

21 1.0 3.0 10.0

22 1.0 4.0 5.0

23 1.0 4.0 7.5

24 1.0 4.0 10.0

25 1.0 5.0 5.0

26 1.0 5.0 7.5

27 1.0 5.0 10.0

Int J Mater Form (2011) 4:389–399 391

adjusted during the learning phase. There are severalalgorithms available among which the Levenberg–Marquardt algorithm (trainlm) will have the fastestconvergence [2]. In many cases, trainlm is able to obtainlower mean square errors than any of the other logrithmstested. This BP network is a multi layer of the networkarchitecture including the input layer, the hidden layer(s)and the output layer. Layers include several processingunits known as neurons. They are connected with eachother by variable weights to be determined. In thenetwork, the input layer receives information fromexternal source and passes this information to the networkfor processing. The hidden layer receives from the inputlayer, and does all information for processing. The outputlayer receives processed information from the network,and sends the results to an external receptor.

Model description

In the development of a multi layer neural network model,several decisions regarding number of neuron(s) in theinput layer, number of hidden layer(s), number of neuron(s)in the hidden layer(s), and number of neuron(s) in theoutput layer and optimum architectures have to bedetermined. Based on the experimental study, the thicknessof the sheet, the size of the perforation, ligament width and

blank width are the highly influenced factors in the formingcharacteristics of perforated Al 8011. Hence these are givenas input parameters to ANN model. The output parametersare the limiting major true strain and limiting minor truestrain. The input and output data set of the model isillustrated schematically in Fig. 4.

Data normalization

In order to provide equal importance of the influence of thesmaller value as like higher valued input variables, both theinput and output variables were normalized within therange −1 to 1 before the training of the network [3, 4]. Thenormalized values (xn) for each raw input/output dataset(di) were calculated as

Xn ¼ 2ðdi � dminÞdmax � dmin

� 1 ð3Þ

Where dmax and dmin are the maximum and the minimumvalues of raw data.

Neural network training

In this study, the total number of data obtained fromexperimental investigation is 189 (27×7=189) amongwhich 126 (two-third) data has been considered for training,

Fig. 2 Punch die assembly

392 Int J Mater Form (2011) 4:389–399

and 69 data for testing. The standard multilayer feedforward backpropagation hierarchical neural networks weredesigned with MATLAB 7.2 Neural Network Toolbox. Thenetworks consist of three layers: the input, the hidden layer,and the output layer. Now, the designed network has fourinput neurons and two output neurons. In the network, eachneuron receives total input from all of the neurons in theproceeding layer as:

netj ¼Xi

W nji ðXiÞn�1 ð4Þ

where netj is the total or net input, Xin is the output of the

node j in the nth layer, and Wjin represents the weights

from node i in the (n-1)th layer to node j in the nth layer. Aneuron in the network produces its input by processing thenet input through an activation (transfer) function which is

usually nonlinear. There are several types of activationfunctions used for BP. However, the tan-sigmoid transferfunction is mostly used which is assigned in hidden layer(s) for processing the inputs as:

f ðxÞ ¼ 2

ð1þ e�xÞ � 1rangeð�1; 1Þ ð5Þ

and purelin a transfer function calculates a hidden layer’soutput from its net input which is assigned for output layer as:

f ðxÞ ¼ ðex � e�xÞðex þ e�xÞ � 1rangeð�1; 1Þ ð6Þ

the weights are dynamically updated using the BPalgorithm. The network has been trained with Levenberg–Marquardt algorithm. This training algorithm has beenselected due to its high accuracy in similar function

Normalize the raw data (input and output)

Select two-third data for training and one third data for testing

Design Network type, training function, Perform function,Number of layers, Neurons and transfer function

Set training parameters

Declare failure network did not converge within

Start

Feed the data into Neural Network Tool box (MATLAB 7.2)

Execute network training

Error within

tolerance

Feed the entire data set to that corresponding architecture

Denormalize the test and entire data

Find the correlation coefficient (R)

Is meanR ≥ 0.95

Select optimum architecture

Declare success/savetraining weightsSelect next architecture

No

Yes

Yes

No

Fig. 3 Flow diagram

Int J Mater Form (2011) 4:389–399 393

approximation. Default training parameters available inMATLAB 7.2 has been set for training the dataset. In orderto judge the performance of the network, the average error(MSE) has been calculated as:

Ep ¼Xpp¼1

Xkk¼1

ðdpk � opkÞ2 ð7Þ

where dpk and opk are the desired and calculated output forkth response respectively. The ‘k’ denotes the number ofneuron in output of network and ‘p’ is the total number ofinstances (epochs). Then, the problem of determining theoptimal number of hidden neurons is a crucial one. Severalstructures have to be considered with different numbers ofhidden neurons to determine the best configuration. Aftertraining, it is denormalized and compared with the

1

2

1

2

3

1

2

3

1

2

InputLayer

OutputLayer

HiddenLayer #1

HiddenLayer #2

Thickness of the sheet

Size of Perforation

Ligament width

Blank Width

Limiting Major True Strain

Limiting Minor True Strain

3

4

18 18

Fig. 4 Schematic illustration ofneural network

Fig. 5 Experimental FLD ofperforated Al 8011 sheets of2 mm thickness

394 Int J Mater Form (2011) 4:389–399

experimental data. The denormalized values (xi) for eachraw output dataset was calculated as

xi ¼ ðxn þ 1Þðdmax � dminÞ2

þ dmin ð8Þ

Where, dmax and dmin are the maximum and minimumvalues of raw data. For testing the prediction ability of the

model, prediction error in each output node has beencalculated as follows:

Pr ediction error % ¼ Actual value� predicted value

Actual value� 100

ð9ÞIn order to determine the optimal architecture, 36

different networks with different number of layers and

Fig. 6 Experimental FLD ofperforated Al 8011 sheets of1.6 mm thickness

Fig. 7 Experimental FLD ofperforated Al 8011 sheets of1.0 mm thickness

Int J Mater Form (2011) 4:389–399 395

neurons in the hidden layer have been designed and testedfor perforated Al 8011.

Result and discussion

The main objective of the proposed research scheme is todevelop an expert system for the perforated aluminium sheetthat can predict it’s forming behavior. To start with, in thepresent work, ANN models are developed to predict theforming limit diagram of Al 8011sheets for a chosen range of

size of perforation and ligament width with differentthickness. Preliminary results are discussed in this section.The deformation behavior was evaluated by the major truestrain and minor true strain.

The experimentally evaluated forming limit diagram forthe perforated Al 8011 sheets of thickness 2 mm, 1.6 and1 mm are shown in Figs. 5, 6 and 7. It is identified that theforming limit strain values are increased with the thicknessirrespective of the size of the perforation and ligament widthand also, the forming limiting stains of the perforated Al8011 sheets are mainly depends with geometrical feature of

Table 3 Correlation coefficient between the network predictions and the experimental values using the test dataset for different network andtraining parameters of perforated Al 8011sheets

Network architecture Major true strain Minor true strain MSE Mean correlation

4–2–2 0.67857472 0.600242223 0.013347 0.639408472

4–4–2 0.72894769 0.707207215 0.00954398 0.718077455

4–6–2 0.82452314 0.676773073 0.00906697 0.750648109

4–8–2 0.84054781 0.721592569 0.00695918 0.781070194

4–10–2 0.73103189 0.632785102 0.00529699 0.681908499

4–12–2 0.80901406 0.728100642 0.00467632 0.768557351

4–14–2 0.96433418 0.886511118 0.00656268 0.925422653

4–16–2 0.93829587 0.650179963 0.00243 0.79423792

4–18–2 0.95205956 0.815928533 0.00241699 0.88399405

4–20–2 0.95011678 0.849101022 0.00146896 0.899608901

4–22–2 0.93915114 0.805663783 0.000992886 0.872407462

4–24–2 0.93915114 0.805663783 0.0008796 0.872407462

4–26–2 0.93915114 0.805663783 0.000467681 0.872407462

4–28–2 0.93915114 0.805663783 0.000334806 0.872407462

4–30–2 0.93915114 0.805663783 0.00029965 0.872407462

4–32–2 0.93915114 0.805663783 7.28E-05 0.872407462

4–2–2–2 0.97343149 0.926462421 1.79E-31 0.949946959

4–4–4–2 0.97315516 0.631778915 0.006638 0.802467039

4–6–6–2 0.86317205 0.725579484 0.1012 converged at 35 epochs 0.79437577

4–8–8–2 0.96317205 0.785579484 0.0024849 0.87437577

4–10–10–2 0.90151060 0.828540425 0.000882542 0.865025517

4–12–12–2 0.93341880 0.890102199 1.217E-21 converged at 13 epochs 0.911760502

4–14–14–2 0.98391511 0.926462421 1.7E-23 converged at 13 epochs 0.955188768

4–16–16–2 0.95391511 0.952421115 1.21E-22 converged at 13 epochs 0.953168115

4–18–18–2 0.97391511 0.974242111 1.877E-26 converged at 13 0.974078613

4–20–20–2 0.90199878 0.89239725 1.8E-31 converged at 14 epochs 0.897198019

4–22–22–2 0.92670229 0.858252498 1.877E-22 converged at 13 epochs 0.892477395

4–24–24–2 0.84864017 0.82384992 5.72E-25 converged at 13 epochs 0.836245046

4–26–26–2 0.94302757 0.877854984 2.74E-23 converged at 8 epochs 0.910441278

4–28–28–2 0.94268441 0.829637916 4.18e-27 converged at 8 epochs 0.886161165

4–30–30–2 0.93577391 0.902826211 2.4E-24 con at 7 0.919300063

4–32–32–2 0.91284222 0.851505064 4.319E-30 con at 7 epochs 0.882173644

4–34–34–2 0.90198283 0.749976155 6.8E-31 con at 7 epochs 0.825979494

4–36–36–2 0.90449405 0.834453054 1.26E-22 con at 6 epochs 0.869473552

4–38–38–2 0.91210095 0.842419458 8.98E-31 con at 7 epochs 0.877260206

4–40–40–2 0.90879691 0.764562453 1.89E-28 con at 6 epochs 0.836679683

396 Int J Mater Form (2011) 4:389–399

the perforated sheet (size of the perforation and ligamentwidth). These strain values are increased with ligament widthand decreased with size of perforation. Forming limit strainsis too difficult to be calculated exactly. However, artificialneural network provides a new way to map the complex andnon-linear relationship. The performance capability of eachnetwork has been examined based on the correlationcoefficient. In order to decide the optimum structure ofneural network, the rate of error convergence was checkedby changing the number of hidden neurons and number ofhidden layers. From Table 3, it is identified that the networkswith two hidden layers of 18 neurons in each layer producedthe best performance for each of the output parameters (4-18-18-2). It has the mean correlation coefficient of 0.97407.

The experimentally evaluated true strains for all the 27samples of Al 8011 in Plane strain region, Tension–Compression region and Tension–Tension region arecompared with the strains predicted from the optimumnetwork (4-18-18-2) of backpropagation neural networkand it is shown in Figs. 8, 9 and 10. From these, it isobserved that there is no significance variation betweenexperimental results and ANN prediction.

Thus, network having two hidden layers of 18 neuronsin each (4-18-18-2), trained with Levenberg-Marquardtalgorithm, has selected as the optimum network. It is alsoobserved that the increase in the number of neurons in thehidden layer had significant improvement on the perfor-mance of the networks and then the performance is

Fig. 8 Comparison of experi-mental results and ANN predic-tion (4-18-18-2) in plane strainregion

Fig. 9 Comparison of experi-mental results and ANNprediction (4-18-18-2) intension–compression region

Int J Mater Form (2011) 4:389–399 397

decreased. The correlation between the predicted values ofthe optimum neural network model and the experimentaldata for prediction of major true strain and minor true strainusing the entire datasets are shown in Figs. 11 and 12respectively.

Conclusions

A study on forming limit diagram of perforated Al 8011sheets by experimental evaluation and artificial neuralnetwork is carried out in this paper. Form the foregoingdiscussion, the following conclusion can be made.

& The perforated Al 8011 sheets having 2 mm thicknessshow higher forming limit strains and having 1 mm

thickness show lower forming limit strains irrespectiveof the geometrical features (size of the perforation andligament width) of the perforated sheet.

& The sheet having 2 mm thickness, 3 mm size ofperforation and 10 mm ligament width possess higherformability and the sheet having 1 mm thickness, 5 mmsize of perforation and 5 mm ligament width possesslower formability in all the mode of plastic deformation,and there is a uniform variation in limit strain ratio.

& The formability of the perforated sheet depends onligament ratio and percentage of open area, and as theligament ratio increases or as percentage of open areadecreases, the formability increases marginally.

Fig. 10 Comparison of experi-mental results and ANN predic-tion (4-18-18-2) in tension–tension region

Fig. 12 Correlation between the predicted values of the neuralnetwork model and the experimental data for prediction of minor truestrain using the entire dataset of Al 8011 sheets

Fig. 11 Correlation between the predicted values of the neuralnetwork model and the experimental data for prediction of major truestrain using the entire dataset of Al 8011 sheets

398 Int J Mater Form (2011) 4:389–399

& Based upon the correlation coefficient and convergence,36 different backpropagation neural network architec-tures are trained/analyzed using the experimental datauntil an optimum architecture is identified.

& Out of the different multi-layer backpropagation neuralnetwork architectures trained, the backpropagationneural network architectures with two hidden layershaving 18 neurons in each (4—18—18—2) trained withLevenberg–Marquardt algorithm was found to be theoptimum network model.

& A sound performance was achieved with the neuralnetwork model, and it shows good correlation betweenthe predicted values of the optimum neural networkmodel and the experimental data for prediction of majortrue strain and minor true strain using the entire datasets

In whole, the formability of the perforated Al 8011sheets from artificial neural network are in good agreementwith the experimental results for chosen thickness of thesheet, size of perforation and ligament width. Also theANN models are able to predict the formability behaviourof the blanks within acceptable error limits.

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