Keywords

1 Introduction

The applications of magnesium alloys have gradually increased due to their high specific strength and good damping properties [1, 2]. These alloys are processed into various useful shapes by forming processes such as forging and extrusion. Extrusion is used to produce tubes, bars, rods, and other solid/hollow profiles at high production rates. The extrusion of magnesium alloys is difficult due to their HCP structure which has a limited number of independent slip systems for plastic deformation and the additional gliding planes slide only at elevated temperatures of more than 225 °C [3] and hence warm extrusion of Mg alloys is preferred. In warm extrusion, the die geometry and the process parameters like extrusion temperature, extrusion ratio, and extrusion speed need to be optimized to minimize the extrusion force. Optimization of the process parameters has been a challenge for many researchers [4,5,6]. In process optimization, simulation of extrusion using the finite element method helps reduce the problems encountered during the extrusion in practice. The thermo-mechanical simulation of Mg alloys is very complex and hence information is scarce. Chandrasekaran and Shyan John [5] studied two-dimensional finite element (FE) simulation of AZ31 alloy which is one of the most commonly used and commercially available Mg alloys. Lapovok et al. [7] used FE simulation to construct an extrusion limit diagram for AZ31 alloy at different extrusion ratios. Hsiang et al. [8] studied the influence of the processing parameters on the extrusion of Mg alloy tubes at elevated temperatures. Hu et al. [9] studied the effect of die geometry on the extrusion of wrought Mg.

Most of the previous works were carried out to develop the optimized die design and the process parameters but very few attempts were made to study the implementation of genetic algorithm (GA) for optimization of process parameters for the Mg alloy extrusion process. In this work, a full factorial technique was used to design the set of simulations for warm extrusion of AZ31 alloy using QForm software, followed by a study of the significance of the process parameters on extrusion load by using ANOVA and the process parameters have been optimized for obtaining minimum peak load using both GA and Minitab Response optimizer tool.

2 Methodology

2.1 Material Selection

The typical composition of AZ31 alloy is given in Table 1.

Table 1 Chemical composition of AZ31 alloy
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The physical properties of the Mg alloy are given in Table 2.

Table 2 Physical properties of AZ31 alloy
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H13 steel is used as the material for the ram and die and its typical composition is given in Table 3. H13 steel is selected as it has a high resistance to thermal cracks (fatigue) due to its excellent toughness in hot conditions.

Table 3 Chemical composition of H13 steel
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3 Simulation of Warm Extrusion

In the present work, the simulation of the extrusion process of AZ31 alloy was carried out by QForm FE software. The design parameters used in extrusion simulation are given in Table 4.

Table 4 Design parameters used in extrusion simulation
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The tool drive conditions were assigned in such a way that the ram could move in the same direction as that of the billet during deformation and the die-container assembly remains fixed. The stopping conditions are implemented in such a way that the simulation stops until the minimum distance between the ram and the die is zero. A hydraulic press with a maximum capacity of 100MN was selected. The material model and friction model used in the simulations are discussed in the following sub-sections. A schematic of the direct extrusion process is shown in Fig. 1.

Fig. 1
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Schematic representation of forward extrusion process of AZ31 alloy

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Material Model. The flow stress values were obtained from the QForm database for a strain range of 0.002 to 3 at strain rates of 0.01, 0.1, 1, 3, and 20 s−1 and temperatures of 300, 350, 400, 450 °C. Hansel Spittel model was used for flow stress dependence on the strain, strain rate, and temperature as shown in Eq. (1).

$$ \sigma \, = \,A e^{m1T} T^{m9} \varepsilon^{m2} e^{m4/\varepsilon } \left( {1 + \varepsilon } \right)^{m5T} e^{m7\varepsilon } \dot{\varepsilon }^{m3} \dot{\varepsilon }^{m8T} $$
(1)

where σ = Stress in MPa, ε = Strain, \(\dot{\varepsilon }\) = Strain rate in s−1, T = temperature in °C at which the test was done. The material constants A, m1, m2, m3, m4, m5, m7, m8, and m9 were calculated using the linear regression method in Minitab software and the values are presented in Table 5. The constants were then used to define the material model in extrusion simulations.

Table 5 Material constants of Hansel–Spittel model
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Considering the material to be isotropic, the Von Mises yield criterion was used to define the yielding behavior of the material.

Friction model. Siebel’s law was used to define friction at the tool workpiece interface. According to the Siebel friction model, shear stresses τ (in MPa) on the contacting surface of a workpiece depend only on the friction factor m, and flow stress σ0 (in MPa) of the workpiece material as given below.

$$\tau =m.\frac{\sigma 0}{\sqrt{3}}$$
(2)

The friction factor m was considered as 0.4 which indicates dry frictional conditions with well-cleaned surfaces [10].

Meshing. The workpiece and the tools were meshed using triangular elements. For the workpiece, mesh adaptation parameter which is the ratio of maximum element size to the size of any element in the mesh was defined. The maximum element size is the normalized size that the solver determines based on the dimensions and shape of a solid. The minimum and the maximum adaptation parameters were considered as 1 and 15, respectively. In a simulation with extrusion temperature of 400 °C, extrusion speed of 0.5 mm/s and semi-die angle of 45°, the maximum and the minimum element size before the simulation were 5.68 mm and 0.14 mm, respectively whereas after the simulation, they were 2.52 mm and 0.13 mm, respectively. The variation of element size of the workpiece during the simulation at a ram displacement of 10 mm is shown in Fig. 2.

Fig. 2
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Simulation result showing variation of element size of the workpiece at a ram displacement of 10 mm

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4 Variation of Warm Extrusion Process Parameters

Usually, in an extrusion process, process parameters like extrusion speed, working temperature, and die angle are of major concern. In the present work on the warm extrusion of AZ31 alloy, the effect of these three process parameters on extrusion load was studied through numerical simulations. The variation of parameters in the simulation is shown in Table 6

Table 6 Levels of process parameters
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The process parameters were varied with full factorial design in the simulations.

5 Results and Discussion

5.1 FEM Simulations

The results obtained from the simulation of direct extrusion of AZ31 alloy in a warm temperature range are described in this section. Figure 3 shows the arrangement of tools (die, container, and ram) and the billet in the FE model. The stress and strain distributions of the workpiece at a ram displacement of 10 mm are shown in Fig. 4a and b respectively. The peak load required for extrusion in each case was predicted from the load v/s displacement graph as shown in Fig. 5. The minimum load obtained from simulations was 1.23 × 105 N at an extrusion temperature of 400 °C, a semi-die angle of 45°, and an extrusion speed of 0.5 mm/s.

Fig. 3
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Simulation of the extrusion process indicating the arrangement of tools (die container and ram) and the billet

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Fig. 4
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Simulation results showing a stress distribution and b strain distribution at a ram displacement of 10 mm

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Fig. 5
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Variation of extrusion load with displacement

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Further analyses were done using ANOVA and GA to identify the significance of each of the extrusion parameters and optimize the process parameters to minimize the extrusion force.

6 Developing Empirical Relationships

To incorporate the effect of the three process variables (extrusion temperature, extrusion speed, and semi-die angles) and their interaction on the peak extrusion load, the polynomial function can be written as

$$\begin{aligned}{\text{Y}}&=\mathrm{ a}0 +{\text{a}}1({\mathrm{A}}) +{\text{a}}2({\mathrm{B}}) +{\text{a}}3({\mathrm{C}}) +{\text{a}}11({\mathrm{A}}2) \\&\quad+{\text{a}}22({\mathrm{B}}2) +{\text{a}}33({\mathrm{C}}2) +{\text{a}}12({\mathrm{AB}}) +{\text{a}}23({\mathrm{BC}})+{\text{ a}}13({\mathrm{AC}})\end{aligned}$$
(3)

where Y is the extrusion load (N), A is the extrusion temperature (°C), B is the semi-die angle (°), and C is the extrusion speed (mm/s). A2, B2, and C2 are the quadratic effects of the process parameters. AB, AC, and BC are the interaction effects between the process parameters. a0, a1, a2, a3, a11, a22, a33, a12, a23, a13 are the coefficients. In the above equation, all the factors may not affect the response significantly. To identify the relative significance of both individual and interaction factors, the ANOVA analysis was used.

Table 7 shows the sum of squares (SS), mean of square (MS), mean square factor to the square of error mean ratio (F), degree of freedom (DF), and the P value to check the fitness of the established model. The model presents \({F}_{\mathrm{0.05,7},46}^{{\text{standard}}}\) = 2.216, for extrusion load, where \({F}^{{\text{regression}}}\) > \({F}_{\mathrm{0.05,7},46}^{{\text{standard}}}\) which confirms that the model is adequate.

Table 7 Analysis of variance
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7 Regression Equation

A confidence level of 95% was considered for framing the regression model so that the significance of each of the factors on the response is high, and the corresponding P value should be less than or equal to 0.05. From Table 7, it is observed that the P value of the model is less than or equal to 0.001, which indicates the fitness of the model is satisfactory. From the F-value in Table 7, it is observed that the extrusion temperature has remarkable linear effects on the extrusion load followed by the semi-die angle. Moreover, the quadratic level of all three parameters has a significant influence on the extrusion load. The Pareto chart clearly shows the contribution of each of the parameters in Fig. 6. The chart shows the absolute values of the standardized effects from the largest effect to the smallest effect. The chart also plots a reference line to indicate the effects which are statistically significant across the reference line which is 2.01. It is observed that the extrusion temperature has the highest contribution to the extrusion load, followed by the quadratic effect of the semi-die angle and extrusion temperature and then the interaction effects of all three process parameters.

Fig. 6
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Pareto chart indicating the contribution of each of the process parameters (linear, quadratic, and interaction effects)

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The developed empirical relationship is shown in Eq. 4.

$$\begin{aligned}{\text{Y}}&= 975786 - 3959\mathrm{A }- 937\mathrm{B }+ 4.742{\text{A}}2 + 18.93{\text{B}}2 \\&\quad- 5267{\text{C}}2 - 2.049\mathrm{AB }+ 50.5{\text{AC}}\end{aligned}$$
(4)

8 Checking the Adequacy of the Relationships

The R2 value indicates the goodness of fit for a model as shown in Table 8. Here, the value of the adjusted R2 is 98.88% which is very high and hence indicates a high significance of the model. The predicted R2 is also in good agreement with the adjusted R2. Moreover, the model prediction ability is commonly measured by the value of predicted R2 which is 98.58% in this case. This indicates that the model provides a good prediction and can be used for experiments in further studies.

Table 8 R2 values
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9 Effect of Process Parameters on Extrusion Load

Figure 7 shows the interaction plot matrix indicating the effect of all three process parameters at various levels of extrusion temperature, extrusion speed, and semi-die angle on peak extrusion load. It is observed that the peak extrusion load decreases with an increase in extrusion temperature which is due to a decrease in the flow stress of the material. Moreover, the peak extrusion load increases with an increase in extrusion speed which is due to an increase in the flow stress of the material. It is also observed that the peak extrusion load is higher at low semi-die angles which is due to high frictional forces at the billet and tool interface and higher at high semi-die angles due to the formation of the dead metal zone at the end of the extrusion process. However, there is an optimum die angle where the extrusion load is minimum.

Fig. 7
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Interaction plot matrix indicating the effect of extrusion parameters on peak extrusion load

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Figure 8a and b show the dead metal zone in the die region for extrusion dies with a semi-die angle of 60° and 75° respectively. It is observed that a higher volume of dead metal in the die region increases the load at the end of the extrusion process due to high die angles. Moreover, extrusion defects like piping defect are likely to occur at higher die angles as shown in Fig. 8b. Hence, in the simulations, it was observed that the extrusion load is higher for dies with higher semi-die angles which is due to a high volume of dead metal at the end of the extrusion process.

Fig. 8
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Sectional view of the dead metal zone in the die region for a extrusion die with 60° semi-die angle b extrusion die with 75° semi-die angle

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10 Optimization of Process Parameters by Genetic Algorithm and Minitab Response Optimizer

To achieve minimum extrusion load, the extrusion parameters were optimized based on the results obtained from simulations by GA and Minitab response optimizer tool.

MATLAB17a program using a “genetic algorithm toolbox” was used to optimize the process parameters. For minimizing the extrusion peak load, Eq. 4 was used as the fitness function. The boundary values used for process parameters are as follows: extrusion temperature between 300 and 400 °C, semi-die angle between 15° and 75°, and extrusion speed between 0.5 and 1 mm/s. The genetic operators, namely, mutation and crossover were used with a probability of 0.2 and 0.6 respectively. Similar conditions were considered by Jurkovic et al. for the optimization of extrusion parameters [11]. After 98 iterations/generations and for a population size of 50, the optimized values for minimizing the extrusion peak load are observed to be an extrusion temperature of 398.67 °C, semi-die angle of 46.186° and extrusion speed of 0.504 mm/s for a peak extrusion load of 1.129 × 105 N.

Minitab response optimizer was also used to compare the optimized results obtained by GA as shown in Fig. 9. Here, the optimum process parameters at which the peak extrusion load is minimum are an extrusion temperature of 400 °C, semi-die angle of 46.515°, and an extrusion speed of 0.5 mm/s for a peak extrusion load of 1.186 × 105 N. It is observed that the optimum process parameter values predicted by GA are also close to that predicted by the Minitab response optimizer.

Fig. 9
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Results obtained from the MiniTab response optimizer

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From the optimum solutions obtained from both the Minitab response optimizer and GA, it can be concluded that the optimum process parameters for warm extrusion of this alloy for industrial applications are an extrusion temperature of 400 °C, a semi-die angle of 45°, and an extrusion speed of 0.5 mm/s. The utility of FE simulations to identify the optimal process variables for warm extrusion has been demonstrated in this work.

11 Conclusion

The warm extrusion of AZ31 alloy was studied using FE simulations by QForm software, and the simulation results were analyzed using Minitab statistical software package. The process parameters (extrusion temperature, semi-die angle, and extrusion speed) were varied at various levels using full factorial design. ANOVA analysis was done to study the significance of each of the process parameters on the extrusion load. Both GA and Minitab response optimizer tools were employed to obtain the optimum process parameters at which the peak extrusion load is minimum. The summary of the work done is discussed below.

  • ANOVA analysis indicates that the linear effect of extrusion temperature has the highest contribution to the extrusion load, followed by the quadratic effect of semi-die angle and extrusion temperature. Although the interaction effect of each of the process parameters is relatively less, their contribution cannot be ignored.

  • GA optimization tool predicts that the optimum process parameters are an extrusion temperature of 398.67 °C, extrusion speed of 0.504 mm/sec, and semi-die angle of 46.186° for a minimum peak extrusion load of 1.129 × 105 N.

  • The Minitab response optimizer tool predicts that the optimum process parameters are an extrusion temperature of 400 °C, extrusion speed of 0.5 mm/sec, and semi-die angle of 46.515° for a minimum peak extrusion load of 1.186 × 105 N. The optimum values predicted by the response optimizer tool are close to those predicted by the GA optimization tool.

  • From the optimum solutions obtained from both the Minitab response optimizer and GA, it can be concluded that the optimum process parameters for warm extrusion of this alloy for industrial applications are an extrusion temperature of 400 °C, a semi-die angle of 45°, and an extrusion speed of 0.5 mm/s.

12 Future Scope

The present paper is focused only on the optimization of process parameters for minimization of peak load during warm extrusion of AZ31 alloy using FE simulations and optimization tools. Since the extrusion ratio considered in the simulations is very small, which is 10:1, the AZ31 alloy may not reach the fracture limit during the process. The prediction of failure using a damage model in the simulations will be carried out in future. Also, the simulation results need to be validated through experiments and then the efficacy of the optimization techniques can be established accurately.