ANN constitutive model for high strain-rate deformation of Al 7075-T6

Journal of Materials Processing Technology 186 (2007) 339–345

ANN constitutive model for high strain-rate deformation of Al 7075-T6 Jamal Sheikh-Ahmad ∗ , Janet Twomey Department of Industrial and Manufacturing Engineering, Wichita State University, Wichita, KS, USA Received 4 August 2006; received in revised form 18 October 2006; accepted 16 November 2006

Abstract An artificial neural network (ANN) constitutive model was developed for Al 7075-T6 based on flow data found in the literature and orthogonal machining tests. The use of orthogonal machining data allowed the ANN network to be trained and tested at high strain-rates of deformation common in machining operations. A new ANN method of network construction (training and validation) was successfully applied to the sparse high strain-rate regime. The method of training and validation, 0.632e stop training method, requires less experimentation to determine network parameters and makes the most efficient use of scarce data. The ANN predictions at high strain-rates where compared with and shown to be superior to a parametric constitutive model. © 2007 Elsevier B.V. All rights reserved. Keywords: Constitutive model; Neural networks; 0.632e error; Al 7075; High strain-rate

1. Introduction Constitutive models are a collection of mathematical equations that describe the macroscopic response of a material to applied stress under different combinations of strain, strain-rate, and temperature. Material constitutive models are widely used in the analysis of manufacturing processes such as metal forming and machining. The accuracy of such analyses greatly depends on the accuracy of constitutive models used. In conventional constitutive theories, a mathematical model is constructed, primarily to represent the behavior of the material at moderate ranges of temperature and strain-rate [1]. The analytical approach to material modeling mainly consists of two aspects; the mathematical formulation of constitutive equations and the determination of material parameters. The former involves the use of principles of mathematics and continuum mechanics, and the latter usually relies heavily on the rational identification and determination of material parameters through analysis of experiments performed on the material. Mathematical models of alloys with various levels of sophistication can be developed depending on the nature of the material behavior and the availability of computing power. Material response predic-

∗ Corresponding author at: The Petroleum Institute, Department of Mechanical Engineering, P.O. Box 2533, Abu Dhabi, United Arab Emirates. Tel.: +971 2 508 5338; fax: +971 2 508 5200. E-mail address: [email protected] (J. Sheikh-Ahmad).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.11.228

tion has an inherent requirement for a high degree of accuracy. Thus the complexity of the mathematical constitutive model corresponds to the degree of accuracy required. This need for more accuracy almost always causes increased complexity in material parameter determination and computations. More complex behaviors such as work hardening and thermal softening are also to be incorporated into the model developed [2]. Apparently, as more material parameters need to be determined, the computation process becomes cost prohibitive. In addition to the increased expense of computation, complexity in mathematical constitutive models makes them difficult to implement. Moreover, they are susceptible to the violation of fundamental principles. For example, a valid material model should satisfy the principles of thermodynamics, the requirements for symmetry and frame indifference [3,4]. Thus this approach to material constitutive modeling is deemed too complex, inaccurate or cost prohibitive to be feasible in most cases. As an alternative to the traditional approach to constitutive model development there is growing interest in artificial neural networks (ANN) as a paradigm of computational knowledge representation. An ANN approach has a number of advantages over the traditional approach; the most important being the ability to learn non-linear relationships between input and output state spaces. For this reason no assumptions or knowledge regarding the mathematical or physical properties governing relationship of inputs to outputs is needed. The development of ANN material constitutive models is still in its nascent phases, with most of the work directed to

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non-metals. Ghaboussi et al. [3] have developed a connectionist constitutive model for concrete. The performance of the developed multi-layer perceptron (MLP) model was acceptable for standard loading conditions for the material. The training was based on the availability of data from non-uniform material tests. Ghaboussi and Sidarta [4] had subsequently developed a nested adaptive neural network (NANN) with an auto-progressive training approach. Their auto-progressive training algorithm was used in lieu of an updated Lagrangian method. The success of this work with concrete initiated constitutive research activity with construction materials. Pernot and Lamarque [5] successfully developed their ANN model for construction sand. The ANN approach in this work was focused upon learning explicit or implicit relationships between the state space variables to approximate the material behavior. The ANN model was a standard multi-layered perceptron (MLP) trained by backpropagation using data from axially compressed concrete behavior. During the same period, Furukawa et al. [6] carried out advanced research to develop an ANN model for cement composites. The research resulted in a standard MLP model trained by backpropagation with input nodes for viscoplastic strain, stress and internal variables. The network performed well for cement composites subjected to axial tensile and crush loads. A significant contribution towards the modeling of alloys was made by Kong et al. [7]. Their ANN model performed within the acceptable range for dynamic impact properties of austenitic steels. The material of interest for this research is aluminum 7075T6. Al 7075 is comparatively strong for an aluminum alloy and has commonly been used in the manufacture of aircraft fuselages owing to its high strength to weight ratio. Modeling the behavior of aluminum alloys is important in many industrial machining processes. High speed machining involves changes in material response, which are required to be predicted prior to the development of the process map. An accurate constitutive model would facilitate the development of an optimized process plan, error-proofing and subsequent process control through improvements in process analysis using tools such as the finite element method. The accuracy of prediction is important to reduce non-conformances of the machined component owing to stresses induced on the material during the process. A parametric constitutive model of Al 7075-T6 was derived by Lee et al. [8]. However, the strain-rate values for the data used to derive this model are well below the strain-rates of deformation experienced in machining. The majority of ANN material models have been developed using the MLP trained by backpropagation [9,10]. In this research the MLP trained by backpropagation is the basis of the ANN model with the addition of the 0.632e stop training method developed by Maradana and Twomey [11] and Maradana [12]. The 0.632e stop training requires less ANN experimentation in modeling and ensures better generalization under conditions of sparse data. The 0.632e method was applied here because data for training and testing in the high strain-rate deformation range of Al 7075-T6 is very limited. The research presented in this paper will demonstrate the advantages of an ANN approach to material constitutive modeling.

Table 1 Orthogonal machining conditions Parameter

Values

Cutting speed, V (m/s) Feed, ac (mm/rev) Rake angle, α Tube diameter (mm) Tube thickness, b (mm) Initial temperature (◦ C)

1.98, 3.96, 11.87 0.05, 0.10, 0.20 5◦ 76 3.81 27, 80, 100

2. Experimental data ANN are a nonparametric or data driven form of modeling. Data to represent the wide range of strain-rate of deformation used in this work comes from two sources. Data for moderate range where taken from published research conducted by Lee et al. [8]. It is noted that the strain-rate values obtained in this work are well below the strain-rates of deformation experienced in machining. To extend the ANN model capability to high strain-rate deformation, additional flow stress data was obtained experimentally from orthogonal machining of aluminum 7075-T6.

2.1. Lee’s data from SHPB tests Data of the dynamic impact properties of aluminum 7075-T6 were obtained experimentally from Split Hopkinson Compression Bar test (SHPB) by Lee et al. [8]. The tests were conducted at three different strain-rate of ε˙ = 1300, 2400 and 3100 s−1 and four different initial temperatures of 25, 100, 200 and 300 ◦ C. From the data a parametric constitutive model was developed. For convenience, data for this research is generated using that model. The parametric model in [8] is given by Eq. (1). σ = 510.58ε0.068 ε˙ 0.144 (1 − 0.00135T ) MPa

(1)

where ε is strain, ε˙ the strain-rate and T is material’s temperature in Kelvin.

2.2. Experimental data from orthogonal machining test Details of the machining tests and results are given in Ref. [13]. The machining experiments involved turning of aluminum 7075-T6 seamless tubes according to the cutting conditions given in Table 1. In addition to cutting at ambient temperatures, the workpiece was also preheated in order to improve its flow properties and to procure data in a higher temperature/strain-rate range. Care was taken to ensure that the preheating would not change the material matrix of the alloy during the actual machining process. Each experiment was conducted twice to check for repeatability of the acquired data. The results from the machining experiments are given in Table 2. The data for flow stress, strain, strain-rate and shear plane temperature was calculated the usual way using classical theory of metal machining as explained below. A model of chip formation as proposed by Oxley [14] and shown in Fig. 1 was used for the analysis of machining data. It is assumed that material forming the chip undergoes shear deformation as it passes through a triangular shear zone (primary zone) that extends from the cutting tool edge to the chip free surface. The thickness of this shear zone increases linearly with distance x from the cutting edge according to the relationship given in Eq. (2). s = A + Bx

(2)

It is assumed that the plane given by the x-axis at the center of the shear zone is the direction of maximum shear strain-rate and hence also the direction of maximum shear stress. The angle φ this plane makes with the direction of cutting speed is calculated using the following equation (Eq. (3)): tan φ =

(ac /ao ) cos α 1 − (ac /ao ) sin α

(3)

where ac is the undeformed chip thickness, ao the chip thickness (measured from chip samples) and α is the rake angle.

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Table 2 Measured and calculated data from orthogonal machining test V (m/s)

ac (mm)

Ti (◦ C)

Fc (N)

Ft (N)

ao (mm)

ε

ε˙ (103 s−1 )

σ (MPa)

Ts (◦ C)

1.98 1.98 1.98 1.98 1.98 1.98 1.98 3.96 3.96 3.96 3.96 3.96 3.96 7.92 11.87 11.87

0.050 0.100 0.125 0.200 0.050 0.100 0.200 0.050 0.100 0.200 0.050 0.100 0.200 0.050 0.100 0.200

28 28 28 28 100 100 100 28 28 28 80 80 80 28 28 28

256.59 401.77 509.26 631.01 204.43 338.6 575.87 213.23 364.2 603.11 169.54 319.43 488.68 209.47 348.87 506.97

154.45 193.57 240.22 263.33 117.9 165.00 217.07 105.73 147.48 202.47 87.52 131.01 134.91 96.61 119.43 129.27

0.098 0.178 0.231 0.241 0.098 0.178 0.241 0.098 0.178 0.366 0.098 0.178 0.368 0.096 0.151 0.305

0.665 0.630 0.640 0.540 0.665 0.630 0.540 0.665 0.630 0.640 0.665 0.630 0.640 0.655 0.580 0.580

60.00 29.40 19.10 30.80 59.90 29.50 30.80 121.00 59.10 22.70 121.00 58.90 22.50 255.00 271.00 99.80

657.40 572.00 582.00 447.80 534.00 479.90 432.00 592.10 551.40 477.40 463.80 481.50 402.40 598.80 566.60 444.10

151.35 153.84 165.07 133.13 198.00 203.01 197.88 163.03 163.03 172.01 184.64 202.60 199.91 187.03 194.71 174.18

The average shear strain, shear strain-rate and shear stress along the x-axis are given by the following relationships (Eqs. (4)–(6)): cos α 2 sin ϕ cos(ϕ − α)

γ=

γ˙ = C τ=

(4)

vs l

(Fc cos ϕ − Ft sin ϕ) sin ϕ bac

cos α V cos(ϕ − α)

Ts =

(6)

where us = τγ is the shear energy per unit volume of material cut, R the fraction of this energy which leaves the shear zone with the chip, Ti the initial temperature of the workpiece, τ the shear stress on the shear plane, ρ the density and c is specific heat. The fraction of shear energy which leaves the shear plane with the chip is given by the following expression (Eq. (12):

(7)

where Fc is the cutting force, Ft the thrust force and b is the width of cut. The constant C is calculated from Eq. (8) as proposed by Lie et al. [15]. C=

 

B 1 ln l + 2A A



− ln

 B  A

Rus + Ti ρc

(5)

where C is a constant for a particular material, l the length of the shear plane, Vs the shear velocity along the shear plane and is determined by the relationship found in Eq. (7). Vs =

where ψ is the direction of grain elongation in the chip and rn is the tool nose radius. ψ was measured from high magnification pictures of the side of chips collected from the machining experiments. The temperature in the primary shear zone is calculated using the model of Loewen and Shaw [16] according to the expression given in Eq. (11).

(8)



(11)

 κγ −1

R = 1 + 1.328

Vac

where κ = K/ρc is thermal diffusivity and K is thermal conductivity of the workpiece material. Both K and c are temperature dependent and an iterative solution was used to determine Ts . The effective stress, strain and strain-rate were determined from the corresponding values in shear using the Von Mises criterion (Eq. (13)): √

γ ε= √ 3

γ˙ ε˙ = √ 3

where A and B are constants that are determined from the cutting geometry using Eqs. (9) and (10).

σ=

A = 2 tan(ψ)

Results from the calculations using Eqs. (2)–(13) are provided in Table 2.

B=

(9)

rn cos(φ − α) tan((π/4) − (α/2))

Fig. 1. Model of chip formation.

(10)

(12)

3τ

(13)

3. The 0.632e stop training method The traditional method for ANN construction is to take a set data of size n, and divide it into a training and validation set; where n = n1 + n2 , n1 is required for training, and n2 for validating the trained network. The ANN architecture (number of hidden layers and nodes) and training parameters (learning rate, momentum factor, and number of epochs) are chosen to achieve the best performance on both the training and validation sets, and is typically carried out through a large number of trials or designed experiments. In some cases the modeler may choose to use stop training. Stop training is a method where as a network trains it is continually evaluated (tested) until the error on a stop training set begins to rise; i.e. training is stopped before the network overfits the training set and fails to generalize well to the test set. In this case the data of sample size n is divided into

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three sets; where n = n1 + n2 + n3 , n1 is used for training, n2 is used for stop training, and n3 for validating the network. This method and the traditional method of ANN construction require significant amounts of data for training and validation. The 0.632e stop training method was developed with the objective to meet the need for a non-linear form of modeling (ANN) requiring a minimum of input or experimentation by the modeler and to make the most effective use of data, especially in cases when data is sparse. The results of the application of the 0.632e method to Al 7075 constitutive modeling demonstrate progress toward this objective. According to the 0.632e method a network of what can be assumed to be a larger than necessary number of hidden nodes is designated. Experimentation is performed to determine learning rate and momentum factor. Like the stop training method, training is stopped before the network overfits. Instead of splitting data into training, stop training, and validation sets, n is used for training and is resampled for stop training and validation. The error used for stop training is an estimate of Efron’s bootstrap 0.632e estimate of expected error of a prediction rule [17–19]. Some particulars of the 0.632e estimate are provided here. Assume that there is some population F from which a random sample of size n is taken. Fˆ is the empirical distribution (the sample of n observations) putting equal mass (1/n) on each observation. The bootstrap method is a resampling technique, where B bootstrap data sets are created by resampling Fˆ with replacement. Each bootstrap data set is of size n observations. The 0.632e estimate of error is one of Efron’s variations on the bootstrap error estimate of a prediction rule. Twomey and Smith [20] found that the bootstrap error estimates for an ANN prediction model to be superior to other resampling methods and the traditional method of ANN validation under conditions of sparse data. The equation for 0.632e is given by Eq. (14). 0.632e = 0.368 × e¯ rr + 0.632 × E0

(14)

where e¯ rr is the training set error and E0 is the error of those points not included in the bootstrap sample. The major draw back to computing E0, is the computational expense of training and testing B additional networks, where B ≥ 20. Efron derived a computationally less expensive estimate of 0.632e, referred in this paper as 0.632e , given by Eq. (15). 0.632e = 0.368 × e¯ rr + 0.632 × ErrGVC(2)

(15)

where ErrGVC(2) is the error estimate given by the groupcross validation method. Group-cross validation is a resampling method without replacement [21,22]. ErrGVC(2) is derived from the error of only two additional networks. Twomey and Smith [20] demonstrated 0.632e to perform approximately equivalent to 0.632e error estimate for an ANN prediction model. According to the 0.632e method, training and validating a network proceeds as follows. The number of input, output and hidden nodes of the application network (the network which will ultimately be applied as the model) is designated. The best training parameters (learning rate and momentum) are determined experimentally. The application network is trained on all n observations. Two other networks, validation 1 and 2 networks,

with architecture and training parameters identical to the application network are created. The n observations are randomly divided into two equal sets n1 and n2 . Validation networks 1 and 2 are trained on data sets n1 and n2 , respectively. The application network and validation 1 and 2 networks train simultaneously. 0.632e is continually computed according to Eq. (14). As stated earlier e¯ rr is the application network error computed on n observations. ErrGVC(2) is derived according to Eq. (16). val err1 + val err2 (16) 2 where val err1 is validation network 1 error computed on data set n2 , and val err2 is validation network 2 error computed on data set n1 . All networks continue to train until 0.632e begins to rise. Since 0.632e was shown by Twomey and Smith [20] to be a good estimate of an ANN prediction model the final 0.632e value is the estimate of the expected true error of the model. Therefore, no additional validation on an independent test set is performed. ErrGVC(2) =

4. ANN constitutive model development A constitutive model is a relationship that describes the macroscopic response of a material under bulk deformation conditions of strain, ε, strain-rate, ε˙ , and temperature, T, and it is usually of the form given in Eq. (17). σ = f (ε, ε˙ , T )

(17)

where σ is the flow stress. From Eq. (17) a simple ANN architecture is created; where the strain, strain-rate and temperature are used as the input vector of the ANN, and the output or predicted value is the flow stress. Unlike many of constitutive material models found in the literature, no special ANN architectures are applied here. The data set representing the behavior of Al 7075-T6 moderate strain-rates [8] consisted of a total of nL = 736 points from which nL train = 585 points were used as the training set, and nL test = 151 as the test set. Of the total data collected experimentally from orthogonal machining in the high strain-rate range, nE = 18, the training set consisted of nE train = 12 data points and the test set consisted of nE test = 6 data points. Combining Lee’s data with the machining data results in the total number of training points of ntrain = 597, and the total number of test points of ntest = 157. In selecting specific observations to include in the test set, a stratified sampling plan was adopted to ensure that the test set was representative of the entire range of values. The distribution of training and test data is provided in Table 3. Note that while 0.632e method of training provides a good estimate of the expected error of the ANN prediction model, a separate test set was used in this research because this is the first real world application of the 0.632e method requiring an independent assessment. A single network of three input nodes representing the three continuous valued inputs strain, ε, strain-rate, ε˙ , and temperature, T, and one continuous valued output, σ, was constructed. Because of the large gap in the range of values between Lee’s data the data of collected experimentally, the data was normal-

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Table 3 Distribution of training and test data Temperature (K)

Lee’s data ε

dε/dt

σ (MPa)

0.005–0.200 0.005–0.200 0.005–0.200 0.005–0.185 0.005–0.200 0.005–0.290 0.005–0.290 0.005–0.275 0.005–0.290 0.005–0.290 0.005–0.290 0.005–0.275

1300 1300 1300 1300 2400 2400 2400 2400 3100 3100 3100 3100

226.5–291.0 361.5–464.5 496.5–638.0 247.4–764.1 247.4–314.4 394.8–519.7 542.3–713.9 652.9–857.4 256.6–338.6 409.6–539.9 562.6–741.5 677.4–889.6

573 473 373 298 573 473 373 298 573 473 373 298

Training nL train

Total Temperature (K)

414.3–486.7

Testing nL test

40 40 39 34 34 58 58 55 59 58 55 55

12 13 14 14 14 12 10 12 13 13 11 13

585

151

Machining data ε

dε/dt

σ (MPa)

Training nE train

Testing nE test

0.54–0.67

4,889–236,982

402.4–657.4

12

6

Table 4 Mean absolute error (S.D.) results of training and testing

Training data Test data

Fig. 2. Plot of 0.632e , e¯ rr, val err1 , and val err2 vs. training iteration number.

ized using the mean and standard deviation. One hidden layer of 20 nodes was chosen and held constant. The learning rate of 0.01 and momentum factor, 0.60, where chosen experimentally to achieve the lowest 0.632e . The number of training iterations was set 30,000. Fig. 2 plots 0.632e , e¯ rr, val err1 , and val err2 by training iteration number. As expected the training set error, e¯ rr, continues to decrease as the number of training iterations increases. At the start of training 0.632e falls, stabilizes after 1200 iterations and begins to rise after 1900 iterations indicating overfit and poor generalization. Training is therefore terminated after 1910 iterations.

Lee’s data

Machining data

Total

1.47 (2.15) 8.17 (6.00)

48.42 (35.90) 49.56 (52.21)

17.34 (8.32) 21.25 (13.48)

occurs for several reasons: (1) the amount of machining training data is substantially less than that of Lee’s data; (2) the machining data contains noisy elements where Lee’s data does not; and (3) although all data was normalized, the machining data is still far outside the range of Lee’s data. The results of the ANN predictions of Lee’s test data are depicted in Fig. 3 and indicate excellent generalization and no problems with overfit. Each of the plots in the figure represents true stress against true strain for a given pair of strain-rate and initial temperature. The curves attest that the neural operator has accurately reproduced the nature of the curve and the ANN model has acquired the capability to approximate the

5. ANN results The performance results of the ANN constitutive model of Al 7075-T6 using the 0.632e method of construction are provided in Table 4. As anticipated the ANN performed very well on Lee’s training and testing data, exhibiting low mean absolute error (MAE) with low standard deviation. A somewhat lower prediction performance is given on the machining data. This

Fig. 3. ANN predictions of Lee’s test data set.

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prediction bias downward. If the machining test data is solely considered, the ANN predictions of flow stress are mostly biased downward (Fig. 4). At the 1920th iteration network training was terminated with a 0.0632e = 0.018. The mean squared error of the test set is MSE = 0.014. (Both values are given in terms of transformed data.) Based on these results the 0.0632e error estimate performs fairly well with an upward bias (∼27% higher). However this is not a fair assessment of the 0.0632e estimate since the test set itself is an estimate the population error. 6. Conclusions Fig. 4. Lee’s model predictions and ANN predictions of machining training and test data sets. Table 5 Mean absolute error (S.D.) of model predictions of machining data

Training data Test data

Lee’s model

ANN model

300.38 (85.5) 253.8 (69.7)

48.42 (35.90) 49.56 (52.21)

increase in true stress with a corresponding decrease in temperature and increase in strain-rate. The ANN constitutive model is also compared to Lee’s parametric constitutive model in the high strain-rate data range experienced in machining. The same model used to generate Lee’s data for training and testing was used to generate predictions of flow stress for the machining conditions given in Table 2. The parametric model predictions are plotted in Fig. 4 along with ANN predictions of machining data, and the MAE of the parametric model predictions are provided in Table 5. Both figure and table reveal the excellent performance of the ANN model as compared to the parametric model in predicting flow stress at high strain-rates. Fig. 4 shows that predictions of the parametric model overestimate the flow stress at high strain-rates encountered in machining. A possible reason for this overestimate is that the mathematical model parameters were obtained from data at much lower strain-rates that they are no longer valid at high strain-rates. The relative frequency histogram of ANN prediction errors (error = target − ANN prediction) over all test data shown in Fig. 5 has an approximate normal distribution indicating slight

The objective of the research presented in this paper was the development of an ANN constitutive model for high strain-rate deformation in Al 7075-T6. This is the first constitutive model of Al 7075-T6 at high strain-rates and the first ANN model of the same. The ANN with the addition the 0.632e method of training and validation was shown to provide highly accurate predictions of values of low strain-rate and very good predictions of values at the more meaningful high ranges of strain-rate. The ANN method of constitutive modeling is successful because it is a non-linear form of modeling that learns the mapping of inputs to outputs. The ANN approach to constitutive modeling described in this paper offer several advantages to shorten development time over the traditional mathematical approach. (a) Lee’s mathematically based constitutive model performed poorly when the range of strain-rates was extended. A new mathematical model that included the higher strain-rates would need to be determined. This research has shown that existing constitutive models can be easily extended with experimental data using an ANN. (b) The 0.632e stop training method was shown to shorten ANN development time by removing the need for experimentation to determine the number of hidden nodes. (c) Unlike some of the ANN constitutive models in the literature, a simple ANN architecture of three inputs, one hidden layer of 20 nodes and one output node, was shown to provide excellent predictions. The simple architecture is not dependent on the use of the 0.632e stop training method. Acknowledgement This work was supported in part by a grant from the National Science Foundation: NSF DMII-9733747. References

Fig. 5. Relative frequency distribution of ANN prediction errors.

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