S.Alikri Sni,G.R.Erhimi,H.Vfeenezhd ,A.R.Kini-Rshid

a School of materials and metallurgical engineering,Iran University of science and technology(IUST),Narmak,Tehran,Iran

b Materials and Polymers Engineering Department,Faculty of Engineering,Hakim Sabzevari University,Sabzevar,Iran

c Department of Metallurgical and Materials Engineering,Faculty of Engineering,Ferdowsi university of Mashhad,Mashhad,Iran

Abstract The aim of the present study was to investigate the modeling and prediction of the high temperature fl w characteristics of a cast magnesium(Mg–Al–Ca)alloy by both constitutive equation and ANN model.Toward this end,hot compression experiments were performed in 250–450°C and in strain rates of 0.001–1 s?1.The true stress of alloy was firs and foremost described by the hyperbolic sine function in an Arrhenius-type of constitutive equation taking the effects of strain,strain rate and temperature into account.Predictions indicated that unlike low strain rates and high temperature with dominant DRX activation,in relatively high strain rate and low temperature values,the precision of the models become decreased due to activation of twinning phenomenon.At that moment and for a better evaluation of twinning effect during deformation,a feed-forward back propagation ANN was developed to study the f ow behavior of the investigated alloy.Then,the performance of the two suggested models has been assessed using a statistical criterion.The comparative assessment of the gained results specifie that the well-trained ANN is much more precise and accurate than the constitutive equations in predicting the hot fl w behavior.

Keywords:Hot deformation;Magnesium alloy;Modeling;Twinning;Hyperbolic sine equation;ANN model.

1.Introduction

Different magnesium alloys have exhibited a great prospective to be used as an advantageous light alloy solution for transportation and aerospace industries for which weight reduction is one of the indispensable concerns in materials selection[1,2].However,unfortunate workability of such alloys limits their applications due to poor formability inherent to the crystal structure of magnesium,i.e.,the hexagonal close-packed(HCP)structure[3].Generally,during homogeneous straining of a polycrystalline material,at least f ve independent slip systems have to be activated.Though at room temperature magnesium takes limited number of slip systems[3,4],which is deficien to fulfil the von Mises criterion.In this regard,twinning plays a significan role in the deformation of magnesium alloys at relatively elevated temperatures[5].Hence,the scaled-up industrialization of the Mg part fabrication is reliant on the hot deformation methods to increase the formability of these alloys.For reliable control and fully understand of industrial thermomechanical treatment,the identificatio of load-bearing capability of alloy in different working conditions is decidedly necessitated.This plays a vital role in performing numerical analysis and findin out the optimal hot forming parameters which really affect the concluding microstructure and subsequent mechanical features of the fina products.

Fig.1.Hot deformation f ow curves of studied alloy at different defomation strain rates,a)0.001 s?1,b)0.01s?1,c)0.1s?1 and d)1s?1.

As mentioned,the formation of twinning phenomenon is conductive to the deformation of magnesium alloys.Among various twinning systems,the{1012}twins have been found to come about most often during high temperature deformation of Mg alloys[6,7].Lots of investigations[8–11]were done to recognize the development of twinning and its related consequences on the deformation performance of different magnesium alloys.In alloys which have twining as their dominant deformation phenomena,such steel and brass alloys[12–14],the strain-hardening rate is high and the true stress–strain curve typically displays a special form.In addition to that,the understanding of the meticulous mechanisms of twinning contribution and its influenc on the work hardening rate is quite vague[11].In many researches,the high stain-hardening rate was attributed to the Hall–Petch effect together with the crystallographic texture effect[11,13].Moreover,twinning can affect strain softening trend such as dynamic recrystallization(DRX)kinetics and intra-grain twininduced recrystallization[13–15].

The hot deformation characteristics of alloys can be interpreted considering both work hardening and work softening phenomena such as cell formation,dislocation tangle,twining,dynamic recovery(DRV),dynamic recrystallization(DRX)and grain growth[16].In the case,establishing an accurate and rational relationships between the f ow stress and these metallurgical phenomena is complicated due to high degree of non-linearity and complexity.In this regard,an extensive amount of research works have been conducted to tackle this matter using different exact analytical and numerical models based on empirical data to accurately describe the high temperature f ow behavior of materials[17,18].Each proposed approach has its own advantages and also negative points.For instance,dislocations dynamics-based exact analytical models of hot deformation entails very strong and comprehensive considerate of the involved controlling mechanisms that are hardly possible to be applied in applied norm.On the other hand,numerical solutions(with just relatively acceptable numerical errors)are less f rmly related to the physical concepts,but still substantial physical understanding of continuum plasticity theory is required.In conclusion,some modeling troubles have be taken into account especially while facing with materials with HCP structures.In such materials,different twinning effects worsen the non-linearity of relationship between the f ow stress and deformation variables in any range of temperatures and strain rates in spite of shortcomings of regression method itself[11,19].

Fig.2.Micrograph of hot deformed at various conditions,250°C-0.01s?1(a),250°C-1s?1(b,d)and 300°C-0.01s?1(c).

Inspired by human biological structure,artificia neural network(ANN)is a novel soft computational model or artificia intelligence approach that tries to model the structure and/or functional aspects of an input(s)-output(s)complex paradigm.It entails of an interconnected collection of artificia neurons and routes information using a brain-like learning approach for computation[20].An ANN can acquire knowledge from limited examples and generalizes the perceived patterns in a series of input and output data deprived of any preceding rules about their logics and interrelations.Since the physical information of deformation mechanisms is not involved in the ANN model,it possess the facility to predict hightemperature deformation behavior of engineering materials in a wide range of hot working domain.Consequently,an artifi cial neural network(ANN)is being progressively utilized in modeling and prediction of the thermomechanical treatment of metallic materials[21–23].

The aim of this study is to perform a comparative study on the constitutive equation model and ANN models in terms of their prediction ability and accuracy for the high-temperature deformation behavior of cast magnesium alloy.Hot compression experimental data from in a specifi range of temperatures(250–450°C)and strain rates(0.001–1 s?1),were employed to model and describe the high temperature f ow curves of alloy considering twinning effect and to develop artificia neural network models.Then,their predicting performance for hot deformation behavior of cast magnesium alloy was evaluated using statistical criteria like correlation coefficien(R2).

2.Experimental procedure

Fig.3.Micrograph of hot deformed at strain rate of 0.01s?1 and temperature of 350°C(a)and 450°C(b).

The experimental magnesium was received in as-cast condition,the chemical composition of which is Mg-7.2Al-1.8Ca-0.11Mn-0.01Fe(wt.%).The as-cast billets were homogenized using a two-stage heat treatment according to ref[24].The cylindrical samples were machined from the homogenized alloys with 8mm diameter and 12mm height using wire cut electro-discharge machine.A Zwick Roell Z250 testing machine equipped with a computerized control furnace was used to perform the hot compression tests in a temperature range of 250–450°C and in strain rates of 0.001–1 s?1.For having near-ideal conditions,the samples were coated with Teflo tape and subsequently soaked at the test temperature for 5min prior to hot deformation.The deformed samples were quenched immediately(less than 3 seconds)for retaining the hot working microstructure.Optical microscopy was employed to examine the microstructures,after mechanical polishing and etching in Picric acid solution.

3.Results and discussion

3.1.Hot deformation behavior

Fig.4.The variation of Ln˙εas function ofσ(a)and Lnσ(b)at strain of 0.5 in the investigated alloy.

The hot deformation curves of the investigated alloy at different conditions are shown in Fig.1.It is common to all the cases of DRX dominated materials that after an initial increasing of f ow stress,the stress level drops after reaching a peak which can be related to the onset and development of DRX phenomena.As can be seen,the f ow stress level is decreased by increasing temperature or decreasing strain rate which in fact is the most preferable condition for as-ease propagation of DRX during microstructural evolution[25,26].Scrutinizing the stress curves,it can be found a linear region in lower temperatures before the peak point of the alloy which can be attributed to the twining phenomenon discussed in the following sections.

Fig.5.The variation of Ln(strainrate)as function of Ln(sinh(ασ0.5))(a)and nLn(sinh(ασ0.5))as function of 1000/RT(b)in the studied alloy.

The micrographs of the alloy at the fina strain(strain of 0.6)are shown in Fig.2.At temperature of 250°C and strain rate of 0.01s?1,many twins can be observed within the coarse primary grains and some micro-cracks or voids are seen as well(Fig.2a).An increase in the extent of twins is obtained at 250°C and by increasing strain rate up to 1 s?1(Fig.2-b,d).As can be seen in the microstructure of the alloy(Fig.2-c),there is no sign of severe twinning at 300°C and strain rate of 0.01 s?1.In fact,in addition to reducing the CRSS(Critical Resolved Shear Stress)of the basal slip system as a result of increasing temperature from 250°C to 300°C,the increase in temperature has led to the activation of additional slip systems(pyramid and prismatic)thus causing the alloy deformation to be less prone to affecting by twinning[3].In contrast to 250°C,at 300°C there is no crack or void in the microstructure implying better deformation of the alloy at 300°C.In addition,looking thoroughly at the microstructures,it can be found that while at 300°C the best sites for DRX development are grain boundaries,at 250°C the preferable sites are the twins(Fig.2-d).A matter that can be related to the higher creative and mobility of dislocations within the twins than the grains[27],it is related to more slip systems at them.Consequently,increasing the energy stored in twins causing the twins to be the most susceptible recrystallizing sites.

Fig.6.The variations ofα,n(a)and Q,LnA(b)with true strain based on the 4-order polynomial fi for investigated alloy.

In Fig.3,the micrographs of the alloy microstructure at 350°C and 450°C shows that by increasing temperature,the coarser primary grains are replaced by the newly recrystallized ones and no twinning or any other forming defects take place at the fina strain.It is deduced that increasing temperature cause the new grains to grow coarser and the DRX fraction to be higher.

3.2.Constitutive equations

Hyperbolic sine equations are usually used to predict the variation of f ow stress with deformation parameters(temperature,strain rate and strain)[28]as follow:

Fig.7.The comparison of f ow stress from experimental and prediction based on constitutive equations at different temperatures under the strain rates a)0.001s?1,b)0.01s?1,c)0.1s?1 and d)1s?1.

Rewriting the above equation,one can have:

whereσis fl w stress and A,α(stress coefficient and n(stress exponent)are constants.Also,the relation between temperature and strain rate is described by the Zener–Hollman(Z)parameter in which Q is the activation energy,T is the absolute temperature and˙εis strain rate.To predict fl w stress,firs it is necessary to determine the constants.To do this,the calculations for strain of 0.5 are described in the following.The stress coefficienαis achieved by dividing βto n1obtained respectively from the exponential equation(˙ε=Bexp(βσ))and the power equation(˙ε=Cσn1).Taking natural logarithms from both sides of the equations the followings can be written:

and

Fig.4-a shows the relationship between Ln˙εandσand the relationship between Ln˙εand Lnσcan be seen in Fig.4b.Based on the best trend line,the values ofβand n1are respectively 0.1269 and 8.6946,resultingαto be 0.014659.Taking the natural logarithm from Eq.(1)while εis 0.5:

Considering at the constant temperature in Eq.(5),n is calculated according to Eq.(6)and the activation energy,Q,can be calculated through Eq.(7).

In Fig.5-a,the variation gradient of is drawn against Ln[sinh(ασ0.5)]atε=0.5 and different temperatures and the stress exponent is calculated 5.98 as the average of 5 gradients.The variation graph of Ln[sinh(ασ0.5)]versus 1000/RT is shown in Fig.5-b based on which Q is calculated as 177.83kJ/mol.Substituting the obtained constants in Eq.(6),LnA is 29.791.

As stated earlier,to predict the f ow stress it is essential to know the constants in Eq.(2)at all the strains.To do this,the constantsα,n,Q and LnA are calculated within the strain range of 0.05 to 0.6 by 0.05 steps.The results can be seen in Fig.6,the change in activation energy from about 218 KJ/mol atε=0.15 to 167 KJ/mol atε=0.6 shows that increasing the strain,the energy required for hot working is significantl decreased that can be related to the DRX development with increasing strain.Actually,with development of DRX there create some small,new and low energy grains in the microstructure facilitating the forming process and reducing the f ow stress.Moreover,the variations in n shows that at strains higher thanε=0.3,the constant for the alloy is about 6 and hot working mechanism of the alloy follow the climb-controlled dislocation creep mechanism which has been reported elsewhere for several alloys[29,30].

3.3.Modeling using by hyperbolic sine equation

To describe the relation between strain and the stress calculated based on the hyperbolic sine equation equations,it was assumed that the alloy’s constants follow a 4-order polynomial function of strain.These functions were extracted based on the curves shown in Fig.6 for the constantsα,n,Q and LnA:

Fig.8.The correlation between the experimental fl w stresses and the predicted ones from the developed hyperbolic sine model.

Fig.9.The work hardening rate(θ-ε)and stress(σ-ε)curves at 250°C-0.1s?1.

At any definit strain,the constants of the alloy are calculated based on the above equations and the f ow stress is predicted according to Eq.(2).In Fig.7,the predicted stress values(the dotted line)are compared with those of the primary curves.The results show that at the higher temperatures(400°C and 450°C)and at all the strains,the real and predicted values are very similar.Also,at 350°C and three low-value strain rates(0.001,0.01 and 0.1 s?1),the stress values predicted based on constitutional equations are of good accordance with the real values.However,as pointed in the Fig.7 by arrows,at temperature of 250°C and at four different strain rates there exists a significan difference between the real and predicted stress values.This was also the case at 300°C and at the highest strain rates(0.1,1 s?1)and 350°C and strain rate of 1 s?1.

Fig.8 compares the variations in predicted and real values of stress.As can be seen,at lower stresses there exists a close intimacy between real and predicted values while no significan relation can be seen in higher stress levels.The relation between the experimental and the predicted stresses at both low and high stress levels can be written as follow based on the Equations Eq.(12)and Eq.(13),respectively.

Fig.10.The work hardening rate curves at different temperatures and the strain rates a)0.001s?1,b)0.01s?1,c)0.1s?1 and d)1s?1.

It should be noted that in the above equations,σTrueand σPredictionare respectively real and predicted stresses.Also,the coefficien of determination(R2)at low and high stress levels is 0.9568 and 0.7375,respectively.According to the derived equations,it can be deduced that at a low value of Z(high temperature and slow strain rate)the modeled stress by constitutive equations is very intimate to that measured experimentally.However,the corresponding difference can be very significan at a high value of Z(low temperature and high strain rate).It seems that at high Z values,twinning takes place,causing difficult in the prediction of stress.

To study the twinning phenomenon,a true stress-true strain curve at temperature of 250°C and strain rate of 0.1 s?1is illustrated in Fig.9 along with a curve of work hardening against strain.As can be seen,the work hardening curve which is the second derivation of the stress-strain curve can be divided into four discrete stages.During stage I,work hardening takes place concurrent to the forming thus the value of work hardening is continuously decreased which can be attributed to the onset and development of dynamic recovery(DRV)during this stage.During the stage II which begins at the strain of about 0.05,the work hardening stabilizes and reaches at a constant stress value(760MPa)which might be related to the twinning phenomenon,this phenomena was seen in another researches[8,31].Twinning reactivates the already slipped systems thus changing the hardening regime.In stage III,by increasing the strain level the work hardening level is reduced by development of DRV and the reduction continues till the beginning of the stage IV,during which work softening of the alloy is taken place by the dominant DRX mechanism.

Fig.11.The variation of work hardening rate value at twins domains as function of LnZ.

Fig.10 shows the curves of work hardening-true strain at different conditions.As can be seen,under a constant strain rate decreasing the temperaturethe hardening level correspond to twinning(stage II)is decreased.In fact the higher is the temperature the lower would be the possibility of twinning.Common to all of the obtained values,the slope corresponding to twinning is negative.

The average value of work hardening corresponding to twinning(θTwin)is obtained from Fig.10.In several conditions,it was impossible to derive work hardening of twinning using the curves and in some of the conditions,i.e.,temperatures of 400°C and 450°C at strain rate of 0.001 s?1,θTwinwas proposed negligible.Fig.11 shows the relation between twinning-induced work hardening and Zener–Holman parameter.As can be seen,θTwinis increased by increasing LnZ and the relation between these two parameters is as follows:

Also,the slope of the work hardening curve can be related to stress and strain through:

Which can be rewritten,and the twinning-induced stress is calculated as:

Fig.12.Architecture of ANN used for modelling hot fl w curves of stuided Mg-alloy.

In fact,deformation by twinning of the alloy cause the fl w stress to be increased so that by increasingθTwinn,the twining-induced stress and the f ow stress are increased.It is deductible from Fig.11 thatθTwinnis increased with LnZ(increasing f ow stress).It is the reason why the modeled curves based on hyperbolic sine equations were erroneous as the effect of twinning is not considered by them(Fig.7).More exactly,in high-stress levels(high Z values)where twinning takes place significantl in the alloy,the behavior of the curve is affected by twinning-induced stress causing the fl w stress level to be increased.It is noteworthy that this fact is not a consideration of the model based on constitutional equations resulting in a difference between the measured and modeled stress values(Fig.8).This is while at low stress condition(low Z values)where no or negligible twinning is taking place,the modeled values are of an appropriate accordance with the measured values.

3.3.ANN model for prediction of fl ow stress

As described in detail at the last section,the significan effect of twinning was not taken into account while mathematically interpreting the true stress-true strain curves of investigated alloy by hyperbolic sine equation model.In this regard and for tackling such problem for establishing an accurate and comprehensive model for modeling hot deformation of Mg alloy,an artificia intelligence approach is implemented called ANN.In the current investigation,multilayer perceptron ANN with feed-forward back-propagation learning algorithm was utilized for predicting high temperature characteristics of studied material.The reason of using multilayer neural network is its relatively superior potential for tackling learning problem while facing nonlinearity and complicity during iterative training process.Strains,logarithmic scale of strain rate and deformation temperature were introduced to network structure as model inputs and the output of the neural computation is f ow stress.The reason of implementing log˙εinstead of˙εis because of two reasons which are;keeping network sensitivity to strain rate variation and logarithmic physical dependence of strain rate to stress.A set of experimental data of hot compression tests performed in the temperature range 250–450°C and strain range 0.05–0.6 and strain rate range 0.001–1 s?1were used to establish the soft computational model.Prior to training,all input values were normalized to the range of 0–1.A dataset that is chosen in random sets were used for training and the rest of them were then employed for blind-testing of ANN.For the purpose of having a well-trained back-propagation ANN model,a variety of setting has to be configure in the network structure.Since gradient descent algorithm is working on the learning epochs of back propagation learning mode,a suitable and effective activation function is required to be accommodated in the network hidden layer(s)and also output layer.Regarding this point,a tangent sigmoid and linear transfer function was adopted in hidden layers and output layer,respectively.In addition,two convergence decisive factors including root mean square(RMS)error between the target value and predicted output and number of iterations were set.In this hot deformation prediction,network architecture with one hidden layer which is made up of 7 hidden neurons(Fig.12)presents most favorable design.

Fig.13.The comparison of f ow stress from experimental and prediction based on ANN model at different temperatures under the strain rates a)0.001s?1,b)0.01s?1,c)0.1s?1 and d)1s?1.

Comparisons of ANN predicted fl w stress with experimental ones for testing data is illustrated in Fig.13.As seen clearly,the ANN predicted values can correctly and properly for the through range of processing parameters,i.e.,temperature,strain rate and strain(Fig.14).Statistical evaluations of R2(0.9993)are a sign of this case that the proposed and develop network is with the acceptable potential for predicting high fl w behavior of investigated Mg alloy at all strain in which both DRX and twinning mechanism is active,as the second one was not well-considered in constitutive model.The neural network with the mentioned optimum designation is successfully incorporated to numerically model the hot deformation characteristics of investigated magnesium alloy.Moreover,as indicated in results,which mean that the welltrained ANN has better prediction capability over the strain compensated constitutive model considering all involved deformation mechanism such as slip and twinning.This examination confirm the outstanding function estimation potential of a single layer neural network to simulate the multifaceted and multi-factor dependent hot deformation this Mg alloy.

Fig.14.The correlation between the experimental fl w stresses and the predicted ones from the ANN model.

4.Conclusions

1.An increase in strain rate and decrease in temperature(high Z values),the extent of DRX is reduced trough microstructure but twining effect enhanced.

2.Results indicated that,hyperbolic sine function can well predict the high temperature deformation behavior of alloy at low Z conditions but due to activation of twinning,it is not trustworthy at relatively high Z values.

3.Work hardening of twinning was studied in different deformation condition and it was seen that the involvement of twinning phenomena is enhanced by increasing Z.

4.ANN prdictions proves its outstanding ability for predicting the dependency of hot fl w stress on its deformation parameters in all strain ranges and considering all involved mechanisms.