Modeling of high temperature rheological behavior of AZ61 Mg-alloy using inverse method and ANN

Materials and Design 29 (2008) 1701–1706

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Modeling of high temperature rheological behavior of AZ61 Mg-alloy using inverse method and ANN M. Talebi Anaraki a, M. Sanjari b,*, A. Akbarzadeh a a b

Department of Materials Science and Engineering, Sharif University of Technology (SUT), Azadi Avenue, Tehran, Iran Materials and Energy Research Center, P.O. Box 14155-4777, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 15 June 2007 Accepted 31 March 2008 Available online 11 April 2008 Keywords: AZ 61 magnesium alloy Rheological behavior Hot deformation ANN Inverse method

a b s t r a c t Inverse method and artificial neural network were employed in modeling the rheological behavior of the AZ61 Mg alloy. The hot deformation behavior of these alloys was investigated by compression tests in the temperature range 250–350 °C and strain rate range 0.0005–0.1 s1. Investigation of stress–strain curves and microstructure of the compression specimen illustrate occurrence of dynamic recrystallization. To determining parameters of two suggested constitutive equations global optimization technique, genetic algorithm, was used. The predicted results by inverse method and ANN depicted a good agreement with the experimental data even if the ANN results has shown the best predicted capability. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The greatest advantage of choosing magnesium alloys for engineering designs lies in its low density, which translates into higher specific mechanical properties. Some other factors such as damping characteristics and stable machinability cause more research interest to these alloys [1,2]. These favorable properties can contribute significantly to the aspect of weight savings in the design and construction of automotive and aerospace components, materials handling equipments, portable tools and even sporting goods [3]. However, magnesium alloys always have poor formability and limited ductility at room temperature due to the intrinsic characteristics of HCP structure such as limited slip systems [4–6]. At room temperature, slip in the basal plane and formation of twinning in the {1 0 1 2} and {1 0 1 1} planes are dominant deformation mechanisms in magnesium alloys [6]. Some other slip systems are active at high temperatures. Duo to the low staking fault energy of magnesium alloys, dynamic recrystallization (DRX) at high temperatures occurs. This is followed by forming new grains at the initial grain boundaries and around the twins as the induced dynamic recrystallization [7,8]. Deeper knowledge about the specific material behavior, especially the flow curves and their dependencies on temperature, strain and strain rate have to be carefully obtained prior to all forming experiments. Also, the effects such as work softening have to be taken into account when theoretically describing the forming behavior for simulation purposes. Knowledge of the behavior of the material during deformation is essential * Corresponding author. Tel.: +98 9353 870870; fax: +98 21 66005717. E-mail address: [email protected] (M. Sanjari). 0261-3069/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2008.03.027

for prediction of the final properties. The fact that relations between the flow stress and strain are more complex in dynamic restoration processes, makes the subsequent analysis even more difficult. A number of stress strain relationships can be found for steel and aluminum alloys but there is a lack of understanding of the underlying constitutive equation for magnesium alloys especially at high temperatures. In this study, the relation of the flow stress of the AZ61-magnesium alloy to strain, strain rate and the temperature are analyzed. In order to develop the rheological behavior, inverse method and neural network are utilized and compared. 1.1. Constitutive equation Numerous data presented in the literature are used for assessment of various stress–strain functions [9–11]. The choice of these functions should account for all phenomena essential for an accurate simulation of the hot forming processes [12]. Analyses of experimental stress–strain curves for magnesium alloy duo to low staking fault energy allow the observation of the influence of the competitive phenomena of hardening and restoration [13]. Difficulties in the mathematical description of the stress–strain functions are caused by the necessity of describing the stress–strain curve for a wide range of strain, strain rate and temperatures [14]. An appropriate way to express the combined effect of the temperature and strain rate on the flow strength is use of Zener– Hollomon parameter (Z) [15]: Z ¼ e_ exp

  Qd n ¼ A½sinhðarÞ RT

ð1Þ

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where the Qd is the activation energy for plastic deformation, R is the universal gas constant, T is the temperature and a (= 0.062 MPa1) [16] and n are material constants. Various relations proposed for rheological behavior of materials to illustrate the DRX at high temperature acquire the strain, Z and some coefficient [17]. For given such a function, it is necessary to determine the Qd. In this study the linear regression was employed to enter the initial range of Qd to optimization. 1.2. Inverse method During the last years, inverse analysis has been developed to evaluate the material constant in various models. Determination of the unknown rheological parameters is now reduced to finding a set value which leads to the best fit between experimental measurements and corresponding computed data [18–21]. The inverse analysis is formulated to determine the constitutive coefficients for which the difference between experimental measurement and computed data is minimal. Quantitatively the analysis is reduced to the minimization of an objective (cost) function that can be written in a least square sense as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X 1 2 /ðx1 ; x2 ; :::; xn Þ ¼ t ð2Þ ðDci  Dm i Þ 2 n i¼1 ðDm i Þ where x1, x2, ..., xn are rheological parameters, n is the total number of sampling points of the measurement. The subscript i designated a number of measurements and Dci and Dm i are the calculated and the measurement values at a given i, respectively. For an appropriate constitutive equation, the best constitutive parameters minimize the above sum [17]. 1.3. Optimization procedures The minimization of an objective function is a major component of the parameter estimation method. However, when the number of parameters is large the objective function may have several minimals. And it is necessary to make sure that the minimal obtained from the optimization identifies the global one. This condition is met when the initial estimates of the optimization variables are close enough to the global minimal. Another way is use of genetic algorithm (GA) that is a global optimization method. GAs, which are optimization techniques have been successfully implemented for a wide range of engineering problems [22,23] that invented by Holland in the early 1970s [24]. Traditionally, a GA starts with an initial population of chromosomes generated usually in a random way. Then, by decoding each chromosome into a search space solution, the value of the fitness function is evaluated. Once the whole population is evaluated, a set of genetic operators like crossover and mutation is applied to create new and better populations. The old population is then replaced by the newly generated one and the evolution process is repeated until the satisfaction of a suitable termination condition is reached [25,26]. 1.4. Neural network Artificial neural networks (ANNs) are basically a data-driven black-box model capable of solving highly non-linear complex problems. Among the various kinds of ANNs the back propagation (BP) learning algorithm has become the most popular in engineering applications [27–29]. By providing a network of this type with a set of training data, the network is able to learn and then adjust the interconnection weights between the layers. This process is repeated until the network performs well on the training set and can subsequently be used to classify or predict from the previously unseen data output [30].

In this work, the neural network inputs were comprised of e, e_ and T, while the flow stress was taken as the output obtained from a compression test. To consider the effect of temperature, friction and variation of strain rate, the input data were primarily corrected. 2. Experimental procedure The alloy used in the present investigation is an AZ61 hot rolled magnesium alloy with initial thickness of 12.5 mm. The composition of this alloy is Mg-6.2% Al-1% Zn-0.11% Mn (in wt percent) and the alloy was utilized from mart. The initial blocks were annealed at 420 °C for 30 min. Cylindrical specimens of 11.7 mm in height and 7 mm in diameter were machined from blocks so that the compression axis is aligned with the normal direction. The specimens were deformed at constant strain rates in an Instron mechanical testing machine. Tests were carried out at temperatures in the range of 250–350 °C and strain rates of 0.0005–0.1 s1 using the graphite lubricant. All specimens were quenched in water after deformation. Specimens for optical microscopy were sectioned, cold mounted, polished and then etched in a solution of 10 ml acetic acid, 4.2 g picric acid, 10 ml water and 70 ml ethanol.

3. Results and discussion 3.1. Hot compression test data In order to consider the effect of friction forces, true stress values were modified according to the following relationship: ¼ r

a r 1þ

2 ffiffi p m hr00 3 3

exp

3e

ð3Þ

2

where r0 and h0 are initial radius and height of specimens, respectively, m the friction factor (m = 0.2 [31]), r  the corrected true stress, a and e the true stress and true strain which correspond to the r homogeneous deformation, respectively and calculated as Ph a ¼ 2 r ð4Þ pr0 h0   h e ¼ ln 0 ð5Þ h P and h are current force and height of the specimen [32,33]. Fig. 1 illustrates the stress–strain curves at various temperatures and constant strain rate (0.01 s1). Strain hardening occurs at the first steps of deformation, leading to a peak stress rp (associated with a strain ep), followed by important softening. Similar trends are observed in the whole experimental temperatures and strain rates. At the first stage of deformation, the hardening process is compensated by recovery before the dislocation density reaches

200 T=250°C T=300°C T=350°C

180 160

True stress (MPa)

1702

140 120 100 80 60 40 20 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

True strain Fig. 1. True stress–true strain curves obtained at a strain rate of 0.01 s1 and temperatures 250, 300 and 350 °C.

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its critical value for onset of dynamic recrystallization (DRX). At the second stage, the strain increases to its critical value for DRX before the equilibrium between hardening and recovery of the dislocation structure is reached. Depending on the experimental conditions (high temperature and/or low strain rates), a steady-state flow stress rss is reached, whereas in the other cases, premature fracture of the sample occurs. The experimental conditions affect also the strain ep corresponding to the peak stress rp. The peak strain decreases with increasing temperature or decreasing strain rate. This is confirmed by metallographic observations of the peripheries of cross section of the tested samples. In the investigated range of temperatures and strain rates, all the samples exhibited DRX (Fig. 2), though the fraction of recrystallised grains varies with temperature and strain rate. Fairly equiaxed microstructures are produced and values between 8 and 25 lm for the recrystallised grains are measured. For a given strain rate, this size increases significantly with increasing temperature but for a given temperature, the effect of strain rate is less pronounced, Fig. 2.

7 0.001 0.01 0.1

6

Log[sinh(ασ)]

5 4 3 2 1 0 1.55

1.65

1.75

1.85

1.95

1000/T(ºK) 0 250 300

-0.5

350

With the aim of linear regression, for a given temperature and stress, inclination of the function log [sinh (ar)] gives a linear relationship between the strain rate and the coefficient n, then Q can be calculated from the following formula: Q ¼ RnT p

ð6Þ

where Tp is slope of log [sinh (ar)] versus 1000/T at constant strain and strain rates [17,34]. In Fig. 3, n and Tp can be determined by measured slop of each line. By calculating n and Tp, a range of Q between 140 and 210 KJ/mol obtained for optimization. After minimization, the optimal value 185 KJ/mol was found for Q.

Log(dε/dt)

-1

3.2. Initial estimation of activation energy for plastic deformation

-1.5 -2 -2.5 -3 -3.5 0

1

2

3

4

5

6

7

8

Log[sinh(ασ)] Fig. 3. Calculation of the activation energy Q in the hot deformed compression samples.

Fig. 2. Microstructures of hot compression samples after deformation: (a) Initial structure, (b and c) at 350 °C and strain rate of 0.001 and 0.1 s1, (d, e and f) at 0.01 s1 and temperatures of 250, 300 and 350 °C, respectively.

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From the Eq. (1) and microstructural observations, it is evident that as Z decreases (at higher T and lower e_ ), dynamic recrystallisation occurs where a high degree of misorientation is reached in the substructure during deformation. At low values of Z, the grain boundaries migrate locally in response to the boundary tensions and local dislocation density variations, and become serrated with a wavelength is closely related to the subgrain size [16]. Grain elongation observed in the alloy is the combined effect of deformation at higher strain rates and higher temperatures. With low temperature deformation, the microstructure is usually very inhomogeneous and these regions are frequently suitable sites for recrystallization during subsequent annealing. The softening processes are more activated with increasing temperature and change the deformation behavior. Cross slip and climb of dislocations are responsible for the observed softening [17]. It is seen (Fig. 2) that as the temperature of deformation increases, the deformation become more homogeneous as the effect of increase in the number of operating slip systems [4]. At low temperatures and high strain rates (high Z) work hardening is dominant, whereas at high temperatures and low strain rates (low Z) the dynamic recovery dominates. Therefore, during hot deformation, the microTable 1 Coefficients in stress–strain Eqs. (7) and (8) obtained for the AZ61 magnesium alloy Eq.

structure will be dependent on the Zener–Hollomon parameter. At high temperature deformation duo to discontinuous dynamic recrystallization high angle grain boundaries formed and consist the new DRX grains. Myshlyeave et al. [18] indicated that with increasing temperature and decreasing the strain rate, the role of recovery on microstructured evolution increased. In fact, the substructure that recovered and formed in the twinning, at the intersection of twins and at the vicinity of initial grain boundaries, cause the formation of subgrains. 3.3. Inverse method results Among many equations suggested for describing the flow stress in hot metal forming processes, one of the best relationships which give good results for materials that exhibit DRX is [17]:  1

 ¼ Kem sinh ðA1 ZÞn expðA2 eÞ r ð7Þ where K, m, A1, n and A2 are material constants and Z is given in the Eq. (7). Another equation that recommended for DRX with introducing two exponential terms is [15]:



  1

 ¼ Kem sinh ðA1 ZÞn 1 þ exp A2 dðe  ep Þ  exp A3 dðe  ep Þ r ð8Þ

K (MPa)

m

A1

n

A2

A3

Q (KJ/mol)

where

1173.83 50.51

1.63 0.24

4.8  1011 5.92  1012

1.27 0.51

12.5 5.3

– 1.54

180 185

dðe  ep Þ ¼ e  ep dðe  ep Þ ¼ 0

a

160

exp. (0.001/s) cal. (0.001/s) exp. (0.01/s) cal. (0.01/s) exp. (0.1/s) cal. (0.1/s)

140

100 80 60 40

220 ANN. (0.001/s) exp. (0.001/s) ANN. (0.01/s) exp. (0.01/s) ANN. (0.1/s) exp. (0.1/s)

200 180 160

True stress (MPa)

True stress (MPa)

120

a

for e > ep

for e < ep

140 120 100 80 60 40

20

20 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0

0.35

0

True strain 160

exp.(0.001/s) cal.(0.001/s) exp.(0.01/s) cal.(0.01/s) exp.(0.1/s) cal.(0.1/s)

140 120

True stress (MPa)

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

True strain

100 80 60 40

b

220 exp.(0.001/s) cal.(0.001/s) exp.(0.01/s) cal.(0.01/s) exp.(0.1/s) cal.(0.1/s)

200 180 160

True stress (MPa)

b

0.05

140 120 100 80 60 40

20

20 0

0 0

0.05

0.1

0.15

0.2

0.25

0.3

True strain Fig. 4. Comparison of true stress–true strain curves obtained at 350 °C and various strain rates by Eq. (7) (a) and Eq. (8) (b).

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

True strain Fig. 5. Comparison of predicted and experimental true stress–true strain curves at 300 °C and various strain rates obtained by (a) ANN and (b) Eq. (8).

M.T. Anaraki et al. / Materials and Design 29 (2008) 1701–1706

In this equation ep is the strain corresponding to the peak stress. For finding dependence of the flow stress of the AZ61-magnesium alloy to the strain, strain rate and temperature, these two equations are utilized. To minimize the objective function (Eq. (2)), the genetic algorithm was employed. For making sure that the genetic algorithm finds the global optimization, a wide range of points are selected to increase the diversity of the populations. Table 1 illustrates the coefficients of the Eqs. (7) and (8) obtained by minimization. Fig. 4 shows the experimental and calculated results for these two equations at 350 °C and different strain rates. It can be seen that Eq. (8) has better fitness to the experimental data. At the high strain rate, the predicted stress is closer to the experimental data for Eq. (8). Also, the predicted result after peak strain shows that the difference between the experimental and calculated stresses is smaller than before the peak which indicates that this equation is more suitable for softening part of the curves. 3.4. The neural network results In this research a network with three input (e, e_ and T) and one output (stress) were considered. Before training, all data were normalized within the interval [1, 1] to enable a comparison of the relative significance of each input. A neural network with three neurons in the input layer (each neuron receiving the three inputs of e, e_ and T), three neurons in the first hidden layer, seven neurons in the second hidden layer, and a single neuron in the output layer, proved to be the most suitable for the present study [30]. A hyperbolic tangent sigmoid transfer function was used for neurons in the hidden layers, whereas a linear transfer function used for output

a

250

1705

layer [29]. The network was trained until the mean squared error (MSE) between the data and network output reduced to 103. Fig. 6a shows the predicted flow stress by ANN model versus the measured value for the testing set. Fig. 5 illustrates the predicted results obtained from ANN at 300 °C and various strain rates that is in good agreement with the experimental measurements. 3.5. Comparison between the inverse method and ANN results However, to make a direct comparison between the ANN approach and the use of constitutive equations the same dataset that utilized during the ANN training for testing are employed. Fig. 6 shows a plot of stress values calculated using the ANN and inverse method versus measured stress data. It can be seen that the ANN analysis actually yields more accurate predictions in contrast with the inverse method. Moreover, Fig. 6 shows that the difference between the predicted results from the inverse method at low stress is more than high stress. 4. Conclusion The rheological behavior of AZ61 magnesium alloy under axisymmetrical hot compression test was modeled using inverse method and artificial neural network (ANN). For constitutive equation of this alloy a formula with six independent constant and two exponential terms was used. The genetic algorithm was utilized for optimization of the fitness function. The results show a good agreement between the experimental and predicted stress, especially after peak stress in the softening part. Alternatively, an artificial neural network approach, which does not requires the constitutive formulations, was able to predict certain flow stress data points better than the constitutive model in terms of the objective function value.

Predicted stress (MPa)

200

References 150

100

50

0

0

50

100

150

200

250

200

250

Experimental stress (MPa)

b

250

Predicted stress (MPa)

200

150

100

50

0 0

50

100

150

Experimental stress (MPa) Fig. 6. Comparison between measured and predicted flow stress values given by (a) ANN and (b) equation 8.

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