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Data Analytics-Driven Selection of Die Material in Multimaterial Co-extrusion of Ti-Mg Alloys

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Abstract
Selection of the most suitable material is one of the key decisions to be taken at the design stage of a manufacturing process. Traditional approaches as Ashby maps based on material properties are widely used in the industry. However, in the production of multimaterial components, the criteria for the selection can include antagonistic approaches. The aim of this work is the implementation of a methodology based on the results of process simulations for several materials and classify them by applying an advanced data analytics method based on Machine Learning (ML), in this case the Support Vector Regression (SVR) and Multi-Criteria Decision Making (MCDM) methodolo-gies, in this case Multi-criteria Optimization and Compromise Solution (VIKOR) combined with Entropy weighting methods. In order to do this, a Finite Element Model (FEM) has been built to evaluate the extrusion force and the die wear in a multi-material co-extrusion process of bimetallic Ti6Al4V-AZ31B billets. After applying SVR and VIKOR combined with Entropy weighting methodologies, a comparison has been established based on the material selection and complexity of the methodology used, resulting that material chosen in both methodologies is very similar and MCDM method is easier to implement because there is no need of evaluate the error of the pre-diction model and the time for data preprocessing is less than the time needed in SVR.
Keywords: 
Subject: Computer Science and Mathematics  -   Artificial Intelligence and Machine Learning

1. Introduction

In the recent years and with the raise of the Industry 4.0, simulation and data analytics methodologies have become more relevant due to their capacity of predict results and being more sustainable compared with the traditional approaches. The need of developing lighter materials in aerospace and automotive industry is increasing to improve the fuel efficiency reducing the environmental impact and increasing the payload to be carried out. Multi-material forming has become a solution because of the capacity to reduce the weight by joining dissimilar materials and also to customize the mechanical properties of the final part to fulfil with the in-service requirements.
Because of this, the Mg and Ti alloys specially AZ31B and Ti6Al4V have become more and more popular due to their low density and good specific strength [1] in the case of Magnesium and their combination of excellent mechanical and physical-chemical properties together with very good strength to weight ratio and superior corrosion resistance [2].
Within multi-material forming highlights co-extrusion process to obtain bimetallic billets composed of a cylindrical sleeve and core made of different materials. Some application cases of multi-material forming processes with these two alloys that can be highlighted are the ones performed by Fernández et al. [3,4] who analysed the effect of the different co-extrusion process parameters by Finite Element Analysis (FEM) simulation using Analysis of Variance (ANOVA) to determine the most relevant ones and also investigated the effect of the selection die material on the co-extrusion process of bimetallic cylindrical billets made of magnesium alloy core and titanium alloy sleeve. Other interesting contributions are the ones performed by Negendanka et al. [5] who carried out a study about the diffusion layer formation under different die angle values in a Mg–core and Al–sleeve billet or Gall et al. [6], who studied the co-extrusion of bimetallic Al–Mg billets into hollow profiles by means of Finite Element Method (FEM) simulation together with experiments.
On the other hand, Machine Learning (ML) [7] has been gaining more relevance in the industry as preferred method to forecast results and anticipate problems [8] by means of algorithms based on statistical methods to detect patterns from data. Support Vector Machines (SVM) is one of the most popular supervised learning methods within ML. It was introduced by Vladimir Vapnik [9] in 1995 and its main applications are classification and regression analysis. For this last one is especially interesting the Support Vector Regression (SVR) module implemented within SVM to estimate discrete values and thus predict future results. Some examples of SVR applications in the industry are the prediction of the laser cutting process cost for AISI316L stainless steel [10], prediction of the cutting force and temperature in bone drilling [11], prediction of the drilling force drilling an internal hole in carbon-fiber-reinforced polymer (CFRP) [12] and applied to wear prediction it can be highlighted the research performed by Benkedjouh et al. [13].
Apart from ML, there are other approaches that allow to take decisions in situations where there are several requirements to fulfil in a complex environment and involving large number of variables. Multi-Criteria Decision Making (MCDM) methods based on multi objective optimization are applied to find the compromise solution to the problem. The first MCDM method was applied by Pareto in 1896 [14] with his famous 80/20 principle. Another example is Saaty in 1977 [15], who used multi-criteria models to solve problems with conflicting goals. Several MCDM methods have been developed and applied to support decision-making in different areas such as, manufacturing process selection [16], supply chain managing contract selection [17] and material selection [18]. In this research a combination of VIKOR [19,20] together with Entropy weighting methods [21,22] has been chosen as MCDM methodology to establish the optimum die material selection.
This study develops two methodologies, one based on SVR and the other applying Entropy weighting method together with MCDM VIKOR, for material selection of the die in a multi-material co-extrusion process to obtain bimetallic billets made of Ti6Al4V-AZ31B. Both methodologies and their results are compared to establish which one gives better results for the problem proposed.

2. Materials and Methods

2.1. Materials, Geometrical Dimensions and Process Parameters

In this study a bimetallic billet made of a Ti6Al4V titanium alloy sleeve and AZ31B magnesium alloy core during a co-extrusion process is analysed.
Figure 1 shows the co-extrusion set up with process parameters and initial dimensions.
Main physical and mechanical properties for Ti6Al4V and AZ31B are shown in Table 1:
Chemical compositions for Ti6Al4V and AZ31B are collected in Table 2 and Table 3 respectively:
The material candidates for the die are extracted from Daniel et al. [4] which chemical composition and physical and mechanical properties are shown in Table 4 and Table 5 respectively.
The extrusion process parameters evaluated during this research are the following:
  • Process parameters: Ram speed (mm/s) and temperature (°C).
  • Tooling parameters: Die semi-angle (°), shear friction factor, and extrusion ratio (A0/Af).
  • Geometric parameters: Shape factor (H0/D0) and diameter ratio (D0/d0).
where A0 and Af are the initial and final area of the cross-section of the billet, D0 and d0 the initial external diameter and internal diameter of the sleeve and H0 the initial billet height.

2.2. Finite Element Modeling and Simulation preparation

Commercial software DEFORM3D© (v11.2) [31] was used to perform finite element simulations.
All parts were meshed with 7000 tetrahedral elements and due to the axial symmetry of the process, only one-quarter of the problem was modeled to reduce the computation time and to avoid heavy database files.
Contact condition among the objects of the simulation is defined as follows. Rigid and elastic objects were considered “masters” (those that deform) and the plastic objects were considered “slaves” (those that are deformed). In the case of the sleeve and core interaction, where both objects are plastic, the titanium alloy was defined as the “master” and the magnesium alloy was defined as the “slave”. All materials were assumed to be isotropic throughout the process.
Heat transfer coefficient between sleeve and core and between sleeve and die was set to 11 N/(s·mm·°C), while between extrusion tooling elements and die was set to 5 N/(s·mm·°C). All the objects of the simulation have 0.02 N/(s·mm·°C) heat transfer coefficient with the air.
The exponential model defined by Wen-juan et al. (2012) [32] was used to define the behaviour of AZ31B while Johnson-Cook constitutive equations [33] were used for the definition of stress-strain curves for the Ti6Al4V.

2.2.1. Tool wear model

Archard’s wear model is used to calculate the wear produced on the surface of the die [34,35,36]. This model is based on Equation (1):
W = ∫ K ·   p a · v b H c · d t
where K is the wear coefficient, P is the interface pressure, v is the sliding velocity between die and billet, H is the hardness and a, b and c are experimentally calibrated coefficients.
The commonly taken value for a and b is 1 while for c is 2 in the case of steel alloys.
K = 2 × 10−5.
Taking into account Equation (1) the parameters to evaluate the wear are ram speed and friction as they can influence in the sliding velocity together with temperature because it has a direct influence in the stress-strain curves.

2.3. Support Vector Regression

SVM works by finding a hyperplane in a high-dimensional space that best separates data into different classes. It aims to maximize the margin (the distance between the hyperplane and the nearest data points of each class) while minimizing classification errors. SVM can handle both linear and non-linear classification problems by using various kernel functions. Unlike SVM used for classification tasks, SVR seeks to find a hyperplane that best fits the data points in a continuous space.
SVR [37] gives the flexibility to define how much error is acceptable in our model and will find an appropriate line (or hyperplane in higher dimensions) to fit the data. Therefore, the goal of SVR is to find a function that approximates the relationship between the input variables and a continuous target variable, while minimizing the prediction error.
As it was said before the idea is to minimize the Equation (2), taking into account the constraints of Equations (3), (4) and (5):
1 2   w 2 + C ∑ i = 1 N ( ξ i + ξ i * )
y i − w x i − b ≤ ε + ξ i
w x i + b − y i ≤ ε + ξ i *
ξ i , ξ i * ≥ 0
where ε is the margin of error while ξ is the deviation from ε also called tolerance margin and w is the classification vector. C is known as the regularized parameter.
The prediction error can be calculated in different ways. One of the most representative is the determination factor (R2) which shows the quality of correlation between the real measured data and the value predicted by Equation (6). A more precise correlation will be obtained for the value of the determination factor nearer to 100%.
R 2 = ∑ i = 1 n ( θ i − θ i m e a n ) ( θ ^ i − θ ^ i m e a n ) 2 ∑ i = 1 n ( θ i − θ i m e a n ) ∑ i = 1 n ( θ ^ i − θ ^ i m e a n )
where θ i is the measurement data, θ ^ i is the predicted magnitude in accordance with SVR, θ i m e a n is the mean of the measurement data and θ ^ i m e a n is the mean of the prediction.

2.4. Entropy method

Entropy method [21,22] is classified within the category of objective weighting methods and it is applicable when the data of decision matrix are known. The entropy is a measure of randomness and disorder in the universe.
Starting with the decision matrix D the project outcomes pij are calculated by means of Equation (7).
D = x 11 ⋯ x 1 n ⋮ ⋱ ⋮ x m 1 ⋯ x m n
p i j = x i j ∑ i = 0 m x i j
where n is the number of criteria and m the number of alternatives.
The entropy measure of project outcomes is obtained as it is shown in Equation (8).
E j = − k ∗ ∑ i = 1 m p i j ∗ l n ⁡ ( p i j )
With k = 1/ln (m).
Objective weight-based definition is given by Equation (9).
w j = 1 − E j ∑ j = 1 n ( 1 − E j )

2.5. VIKOR method

VIKOR method [38,39] stands for VIseKriterijumska Optimizacija I Kompromisno Resenje, which means Multi-criteria Optimization and Compromise Solution.
This methodology is based on the concept that the compromise solution is the one which is at minimum distance for the ideal solution and at the same time at maximum distance for the anti-ideal solution. VIKOR request a validation step before declaring the compromise solution feasible.
After the criteria to be evaluated are defined the decision matrix (D) is built.
D = x 11 ⋯ x 1 n ⋮ ⋱ ⋮ x m 1 ⋯ x m n
At this point, the best f b * and worst f b − for each criteria rating values of the decision matrix.
f b * = m a x ⁡ ( x i b ) f b − = m i n ⁡ ( x i b ) Whether the objective is to maximize the criteria.
f b * = m i n ⁡ ( x i b ) f b − = m a x ⁡ ( x i b ) Whether the objective is to minimize the criteria.
Where b = 1 …m being m the number of criteria took into account and i = 1 …n and n is the number of the alternatives.
Utility measure (Sj) and Regret measure (Rj) are calculated according with Equations (10) and (11):
S j = ∑ b = 1 m W b ∗ f b * − f i j f b * − f b −
R j = m a x W b ∗ f b * − f i j f b * − f b −
where Wb are the weight values obtained in the case of this study after applied Entropy weighting methods explained above.
Index Q can be obtained by means of the Equation (12):
Q a = υ ∗ S j − S * S − − S * + 1 − υ ∗ R j − R * R − − R *
where:
S − = m a x ⁡ ( S j )
S * = m i n ⁡ ( S j )
R − = m a x ⁡ ( S j )
R * = m i n ⁡ ( S j )
υ is a parameter which represents the type of voting used during the process (υ > 0.5 means “vote by majority rule”, υ = 0.5 “vote by consensus” and υ < 0.5 “with vote”).
The lowest Qa value indicates the best alternative solution and it can be recommended if the following conditions are satisfied:
The “acceptable advantage” condition means that Q(a’’) – Q(a’) ≥ DQ. Being a’’ the alternative in second position in the ranking list by Qa and a’ the first one. DQ is defined by Equation (13):
D Q = 1 ( n − 1 )
where n is the number or alternatives.
Finally, the “Acceptable stability in decision making” condition implies that a’ alternative must also be the best ranked in Sj and/or Rj. If one of the conditions is not fulfilled, then a set of compromise solutions is proposed.

2.5. Methodology

Two different methodologies have been proposed for the selection of the optimal die material in order to obtain the minimum extrusion force and die wear. The methodology steps are shown in Figure 3 flowchart.
The criteria for the final results comparison are:
➢
Simplicity.
➢
Amount of data from simulations.
➢
Time consuming.
The prediction was carried out in Python software [40].

3. Results

In this paper a set of simulations of a multi-material co-extrusion process have been performed by using commercial software DEFORM3D© (v11.2) followed by application of two different methodologies to choose which is the best die material to obtain minimum extrusion force and minimum wear during the process. For the list of simulations carried out in the present work see Table A1 in the Appendix A.

3.1. SVR Methodology

As explained above the dataset is obtained from Table 6 and for each material and each parameter to be predicted, several dataframes were obtained by using “pandas” together with “sklearn” libraries.
Using “RFE” module for Regression Feature Selection from “sklearn.feature_selection” together with “SVR” module from “sklearn.svm”, the influence of the process parameters are ranked in accordance with their influence in the extrusion force as it is shown in Table 6.
Taking into account these results, it can be said that friction is the most important process parameter while temperature is the less important one. As there is not a clear pattern about the influence of each process parameters and this influence is clearly dependent on the die material, for the prediction model all the parameters will be implemented.
For the prediction model of the extrusion force, the dataframes for each material were split in two groups, one for training and one for testing using the “train_test_split” function from “sklearn.model_selection” module, being the test size 0.3.
After applying the “LinearRegression” function from “sklearn.linear_model” to build the prediction model using the training data and afterwards evaluate the model using the test data, the determination factor (R2) for each material is shown in Table 7:
Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 show the comparison between the simulation obtained values and the prediction ones.
Using the prediction models for each die material a bigger number of results for the extrusion force can be compared without the need of performing more simulations. Table 8 show the ranking of the die materials as function of the times that their prediction value for the extrusion force is the lowest one.
If there was only the minimum extrusion force as requirement for the die material election, AISI3310 would be the chosen one followed by AISI316 and H13 sharing the second position in the ranking.
The SVR methodology is now applied for the wear prediction with the following modification.
Due to the results variation is not possible to apply a linear model regression but a polynomial one. In order to do this is necessary to import “PolynomialFeatures” module from “sklearn.preprocessing” library to generate a new feature matrix consisting of all polynomial combinations of the features with degree less than or equal to the specified degree (in this case a 2 degree polynomial is used).
In Table 9 and Table 10 are shown the process parameters ranking and the determination factor (R2) for the wear model:
The prediction model is not as accurate as the one for the extrusion force. This can be due to the number of simulations performed to obtain the wear distribution are lower than for the extrusion force because of Archad’s wear model only takes into account temperature, friction and ram speed as it was mentioned in paragraph 2.2.1.
Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the comparison between the simulation obtained values and the prediction ones.
Table 11 show the ranking of the die materials as function of the times that their prediction value for the die wear is the lowest one.
Finally, a crosscheck between Table 8 and Table 11 is performed to rank the die material which fulfil better the minimum extrusion force and the minimum die wear as it can be seen in Table 12.
In both rankings AISI3310 is the best choice to reduce the extrusion force and die wear. AISI316 and H13 have the same position for the extrusion force but not for the die wear, this is the reason because in the final ranking AISI316 is positioned better thank H13. The die materials that can be rejected as feasible option are AISI52100 and 25CrMo4.

3.2. MCDM Methodology

As explained in paragraph 3. the dataset is obtained from Table A1. In the MCDM methodology the weights are calculated by means of Entropy method and afterwards VIKOR method is applied to classify the different materials based on criteria rating values decision.
For the Entropy method, the normalized matrix is:
0.1923 0.2079 0.2110 0.1998 0.1968 0.2021 0.1727 0.1947 0.1947 0.1913 0.1925 0.2011 0.1000 0.1842 0.1459 0.2035 0.1993 0.2214
0.1926 0.1903 0.2102 0.2001 0.1972 0.2023 0.1729 0.1954 0.1949 0.1977 0.2311 0.1717 0.1764 0.1829 0.1563 0.2175 0.2105 0.2118
0.2137 0.1952 0.1842 0.1999 0.2011 0.1996 0.2179 0.2008 0.2010 0.1884 0.1978 0.1667 0.1701 0.2078 0.2568 0.1908 0.1881 0.2067
0.1947 0.2046 0.1903 0.2011 0.2024 0.1967 0.2162 0.2034 0.2059 0.2642 0.2233 0.2733 0.3639 0.2532 0.2665 0.2315 0.2407 0.2565
0.2068 0.2019 0.2043 0.1990 0.2025 0.1994 0.2202 0.2057 0.2036 0.1584 0.1553 0.1871 0.1897 0.1719 0.1744 0.1567 0.1614 0.1037
Then, the entropy array (Ej) is calculated:
Ej = [0.999418867 0.999681506 0.999085371 0.999996437 0.999952024 0.999966715 0.996102816 0.999852667 0.999839187 0.990960752 0.99430503 0.989129167 0.944884512 0.993690857 0.979914551 0.994771731 0.99470602 0.976990905]
The weights are presented in Table 13:
In VIKOR the best f b * and worst f b − values for each criterion are obtained directly from decision matrix D.
86.019 136.529 228.511 307.324 97.898 88.621 88.040 84.512 83.759 0.367 0.318 0.302 0.124 0.417 0.253 0.378 0.361 0.370
86.153 124.955 227.644 307.783 98.095 88.706 88.140 84.799 83.850 0.379 0.382 0.257 0.219 0.414 0.271 0.404 0.381 0.354
95.608 128.196 199.579 307.520 100.048 87.505 111.049 87.149 86.484 0.361 0.327 0.250 0.211 0.471 0.445 0.354 0.341 0.345
87.120 134.333 206.128 309.397 100.654 86.238 110.203 88.307 88.602 0.507 0.369 0.410 0.452 0.573 0.462 0.430 0.436 0.428
92.531 132.578 221.350 306.154 100.708 87.422 112.221 89.296 87.588 0.304 0.257 0.281 0.236 0.389 0.303 0.291 0.292 0.173
447.431 656.591 1083.212 1538.178 497.402 438.492 509.653 434.063 430.283 1.918 1.652 1.499 1.242 2.265 1.735 1.856 1.812 1.669
fi* 86.019 124.955 199.579 306.154 97.898 86.238 88.040 84.512 83.759 0.304 0.257 0.250 0.124 0.389 0.253 0.291 0.292 0.173
fi- 95.608 136.529 228.511 309.397 100.708 88.706 112.221 89.296 88.602 0.507 0.382 0.410 0.452 0.573 0.462 0.430 0.436 0.428
Utility measure (Sj) and Regret measure (Rj) are obtained:
Sj Ri
0.23758265 0.12075866
0.36022622 0.11092085
0.44932592 0.1258087
0.98459134 0.37557176
0.21183731 0.12764252
S* 0.21183731 R* 0.11092085
S- 0.98459134 R- 0.37557176
Using the values S*, S-, R* and R- together with the assumption of vote by consensus (υ = 0.5), the index Q is calculated:
Qi
AISI3310 0.03524455
H13 0.09601303
AISI52100 0.18179111
25CrMo4 1
AISI3310 0.03159193
In VIKOR the index Q is ranked from the lowest to the highest value, therefore the best material to obtain minimum extrusion force and minimum die wear is AISI3310. But before recommending this material as best compromise solution the conditions of “Acceptable advantages” and “Acceptable stability in decision making” have to be fulfilled.
In this case DQ = 0.25 according with Equation (13). Then:
Q(2) – Q(1) = 0.0365261
Q(3) – Q(1) = 0.0644211
Q(4) – Q(1) = 0.15019917
Q(5) – Q(1) = 0.96840807 > DQ
Q(1) = S*
As only the second condition is fulfilled, a set of compromise solution is presented and ranked in Table 14

4. Discussion

In this paper two methodologies are proposed to choose the best material for the die in a multi-material coextrusion process, taking into account that the process has to fulfil the requirements of minimum extrusion force and minimum die wear.
The first methodology proposed is the SVR based on SVM. The main advantage is the prediction model obtained during the process which allows the engineers to know the outcomes when varying the process parameters. On the other hand, the disadvantages are the number of simulations needed to obtain a good prediction model and depending on the results of those simulations the complexity to obtain the prediction model can be very high.
MCDM methodology allows to select the best die material with a smaller number of simulations than the SVR one and without considering the accuracy or complexity of prediction models. Also, it is less time consuming because Entropy and VIKOR methods can be applied directly to the data and there is no need to have knowledge in programming languages like Python.
The results for the top three materials selected are the same independently of the methodology applied. Therefore, if there is no need to obtain a prediction model to forecast results by applying other values to the parameters, the die material selection methodology recommended is MCDM one due to its simplicity and time consuming to implement it.
Finally, for future research it would be interested a comparison among different machine learning methods to obtain a more robust prediction model not only for the wear but also for other parameters such as damage factor, mean stresses, microstructure resultant and so on.

Author Contributions

Conceptualization, D.F., A.R.-P., and A.M.C.; methodology, D.F.; formal analysis, D.F., A.R.-P., and A.M.C.; investigation, D.F., A.R.-P., and A.M.C.; resources, A.R.-P. and A.M.C.; writing—original draft preparation, D.F.; writing—review and editing, A.R.-P. and A.M.C.; supervision, A.R.-P. and A.M.C.; project administration, A.R.-P. and A.M.C.; funding acquisition, A.R.-P. and A.M.C. All authors read and agreed to the published version of the manuscript.

Funding

This research was funded within the framework of the “Doctorate Program in Industrial Technologies” of the UNED and it has been funded by the project 2021V/-TAJOV/006 (awarded in the UNED Research Projects call named “Young Talents 2021”).

Data Availability Statement

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Acknowledgments

We would like to extend our acknowledgement to the Research Group of the UNED “Industrial Production and Manufacturing Engineering (IPME)” and the Industrial Research Group “Advanced Failure Prognosis for Engineering Applications”.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Appendix A

Table A1. List of simulations performed by DEFORM3D© (v11.2).
Table A1. List of simulations performed by DEFORM3D© (v11.2).
Simulation Material Ram speed(mm/s) Core
diameter (mm)		 Billet
Height (H)		 Temperature(⁰ C) Friction Die
semi-angle (⁰)		 Extrusion
Ratio
1 AISI316 2 5 20 200 0.2 30 1.78
2 AISI316 2 6 15 100 0.2 30 2.25
3 AISI316 2 7 25 100 0.3 30 1.44
4 AISI316 3 6 15 200 0.3 15 2.25
5 AISI316 3 7 15 300 0.2 45 1.44
6 AISI316 2 6 20 200 0.1 30 1.78
7 AISI316 2 6 20 200 0.1 15 1.78
8 AISI316 2 6 20 200 0.1 45 1.78
9 AISI316 2 6 20 200 0.1 60 1.78
10 AISI316 2 6 20 200 0.1 75 1.78
11 AISI316 2 6 20 200 0.1 90 1.78
12 AISI316 2 2 20 200 0.1 30 1.78
13 AISI316 2 4 20 200 0.1 30 1.78
14 AISI316 2 8 20 200 0.1 30 1.78
15 AISI316 2 10 20 200 0.1 30 1.78
16 AISI316 2 6 15 200 0.1 30 1.78
17 AISI316 2 6 25 200 0.1 30 1.78
18 AISI316 2 6 30 200 0.1 30 1.78
19 AISI316 2 6 35 200 0.1 30 1.78
20 AISI316 2 6 20 200 0.2 30 1.78
21 AISI316 2 6 20 200 0.3 30 1.78
22 AISI316 2 6 20 200 0.4 30 1.78
23 AISI316 2 6 20 200 0.5 30 1.78
24 AISI316 2 6 20 200 0.6 30 1.78
25 AISI316 2 6 20 200 0.7 30 1.78
26 AISI316 2 6 20 100 0.1 30 1.78
27 AISI316 2 6 20 300 0.1 30 1.78
28 AISI316 1 6 20 300 0.1 30 1.78
29 AISI316 3 6 20 300 0.1 30 1.78
30 AISI316 4 6 20 300 0.1 30 1.78
31 AISI316 2 6 20 200 0.1 30 1.44
32 AISI316 2 6 20 200 0.1 30 2.25
33 AISI316 2 6 20 200 0.1 30 2.94
34 H13 2 5 15 100 0.1 15 1.44
35 H13 2 6 25 300 0.1 15 1.78
36 H13 3 5 15 300 0.3 30 1.78
37 H13 3 6 25 100 0.2 45 1.44
38 H13 3 7 25 200 0.1 30 2.25
39 H13 2 6 20 200 0.1 30 1.78
40 H13 2 6 20 200 0.1 15 1.78
41 H13 2 6 20 200 0.1 45 1.78
42 H13 2 6 20 200 0.1 60 1.78
43 H13 2 6 20 200 0.1 75 1.78
44 H13 2 6 20 200 0.1 90 1.78
45 H13 2 2 20 200 0.1 30 1.78
46 H13 2 4 20 200 0.1 30 1.78
47 H13 2 8 20 200 0.1 30 1.78
48 H13 2 10 20 200 0.1 30 1.78
49 H13 2 6 15 200 0.1 30 1.78
50 H13 2 6 25 200 0.1 30 1.78
51 H13 2 6 30 200 0.1 30 1.78
52 H13 2 6 35 200 0.1 30 1.78
53 H13 2 6 20 200 0.2 30 1.78
54 H13 2 6 20 200 0.3 30 1.78
55 H13 2 6 20 200 0.4 30 1.78
56 H13 2 6 20 200 0.5 30 1.78
57 H13 2 6 20 200 0.6 30 1.78
58 H13 2 6 20 200 0.7 30 1.78
59 H13 2 6 20 100 0.1 30 1.78
60 H13 2 6 20 300 0.1 30 1.78
61 H13 1 6 20 300 0.1 30 1.78
62 H13 3 6 20 300 0.1 30 1.78
63 H13 4 6 20 300 0.1 30 1.78
64 H13 2 6 20 200 0.1 30 1.44
65 H13 2 6 20 200 0.1 30 2.25
66 H13 2 6 20 200 0.1 30 2.94
67 AISI52100 2 5 15 100 0.1 15 1.44
68 AISI52100 2 6 25 300 0.1 15 1.78
69 AISI52100 3 5 15 300 0.3 30 1.78
70 AISI52100 3 6 25 100 0.2 45 1.44
71 AISI52100 3 7 25 200 0.1 30 2.25
72 AISI52100 2 6 20 200 0.1 30 1.78
73 AISI52100 2 6 20 200 0.1 15 1.78
74 AISI52100 2 6 20 200 0.1 45 1.78
75 AISI52100 2 6 20 200 0.1 60 1.78
76 AISI52100 2 6 20 200 0.1 75 1.78
77 AISI52100 2 2 20 200 0.1 30 1.78
78 AISI52100 2 4 20 200 0.1 30 1.78
79 AISI52100 2 8 20 200 0.1 30 1.78
80 AISI52100 2 10 20 200 0.1 30 1.78
81 AISI52100 2 6 15 200 0.1 30 1.78
82 AISI52100 2 6 25 200 0.1 30 1.78
83 AISI52100 2 6 30 200 0.1 30 1.78
84 AISI52100 2 6 35 200 0.1 30 1.78
85 AISI52100 2 6 20 200 0.2 30 1.78
86 AISI52100 2 6 20 200 0.3 30 1.78
87 AISI52100 2 6 20 200 0.4 30 1.78
88 AISI52100 2 6 20 200 0.5 30 1.78
89 AISI52100 2 6 20 200 0.6 30 1.78
90 AISI52100 2 6 20 200 0.7 30 1.78
91 AISI52100 2 6 20 100 0.1 30 1.78
92 AISI52100 2 6 20 300 0.1 30 1.78
93 AISI52100 1 6 20 300 0.1 30 1.78
94 AISI52100 3 6 20 300 0.1 30 1.78
95 AISI52100 4 6 20 300 0.1 30 1.78
96 AISI52100 2 6 20 200 0.1 30 1.44
97 AISI52100 2 6 20 200 0.1 30 2.25
98 25CrMo4 2 6 20 200 0.3 45 1.44
99 25CrMo4 2 7 15 200 0.1 45 1.78
100 25CrMo4 3 5 25 200 0.2 15 1.44
101 25CrMo4 3 7 20 100 0.3 15 1.78
102 25CrMo4 2 6 20 300 0.1 30 1.78
103 25CrMo4 2 6 20 200 0.1 30 1.78
104 25CrMo4 2 6 20 200 0.1 15 1.78
105 25CrMo4 2 6 20 200 0.1 45 1.78
106 25CrMo4 2 6 20 200 0.1 60 1.78
107 25CrMo4 2 6 20 200 0.1 75 1.78
108 25CrMo4 2 6 20 200 0.1 90 1.78
109 25CrMo4 2 2 20 200 0.1 30 1.78
110 25CrMo4 2 4 20 200 0.1 30 1.78
111 25CrMo4 2 8 20 200 0.1 30 1.78
112 25CrMo4 2 10 20 200 0.1 30 1.78
113 25CrMo4 2 6 15 200 0.1 30 1.78
114 25CrMo4 2 6 25 200 0.1 30 1.78
115 25CrMo4 2 6 30 200 0.1 30 1.78
116 25CrMo4 2 6 35 200 0.1 30 1.78
117 25CrMo4 2 6 20 200 0.2 30 1.78
118 25CrMo4 2 6 20 200 0.3 30 1.78
119 25CrMo4 2 6 20 200 0.4 30 1.78
120 25CrMo4 2 6 20 200 0.5 30 1.78
121 25CrMo4 2 6 20 200 0.6 30 1.78
122 25CrMo4 2 6 20 200 0.7 30 1.78
123 25CrMo4 2 6 20 100 0.1 30 1.78
124 25CrMo4 2 6 20 300 0.1 30 1.78
125 25CrMo4 1 6 20 300 0.1 30 1.78
126 25CrMo4 3 6 20 300 0.1 30 1.78
127 25CrMo4 4 6 20 300 0.1 30 1.78
128 25CrMo4 2 6 20 200 0.1 30 1.44
129 25CrMo4 2 6 20 200 0.1 30 2.25
130 AISI3310 2 6 20 200 0.1 30 1.78
131 AISI3310 2 6 20 200 0.1 15 1.78
132 AISI3310 2 6 20 200 0.1 45 1.78
133 AISI3310 2 6 20 200 0.1 60 1.78
134 AISI3310 2 6 20 200 0.1 75 1.78
135 AISI3310 2 6 20 200 0.1 90 1.78
136 AISI3310 2 2 20 200 0.1 30 1.78
137 AISI3310 2 4 20 200 0.1 30 1.78
138 AISI3310 2 8 20 200 0.1 30 1.78
139 AISI3310 2 10 20 200 0.1 30 1.78
140 AISI3310 2 6 15 200 0.1 30 1.78
141 AISI3310 2 6 25 200 0.1 30 1.78
142 AISI3310 2 6 30 200 0.1 30 1.78
143 AISI3310 2 6 35 200 0.1 30 1.78
144 AISI3310 2 6 20 200 0.2 30 1.78
145 AISI3310 2 6 20 200 0.3 30 1.78
146 AISI3310 2 6 20 200 0.4 30 1.78
147 AISI3310 2 6 20 200 0.5 30 1.78
148 AISI3310 2 6 20 200 0.6 30 1.78
149 AISI3310 2 6 20 200 0.7 30 1.78
150 AISI3310 2 6 20 100 0.1 30 1.78
151 AISI3310 2 6 20 300 0.1 30 1.78
152 AISI3310 1 6 20 300 0.1 30 1.78
153 AISI3310 3 6 20 300 0.1 30 1.78
154 AISI3310 4 6 20 300 0.1 30 1.78
155 AISI3310 2 6 20 200 0.1 30 1.44
156 AISI3310 2 6 20 200 0.1 30 2.25
157 AISI3310 2 6 20 200 0.1 30 2.94

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Figure 1. Co-extrusion set up with process parameters and initial billet dimensions.
Figure 1. Co-extrusion set up with process parameters and initial billet dimensions.
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Figure 2. 2D hyperplane representation.
Figure 2. 2D hyperplane representation.
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Figure 3. Methodology flowchart.
Figure 3. Methodology flowchart.
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Figure 4. AISI316 extrusion force prediction comparison.
Figure 4. AISI316 extrusion force prediction comparison.
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Figure 5. H13 extrusion force prediction comparison.
Figure 5. H13 extrusion force prediction comparison.
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Figure 6. 25CrMo4 extrusion force prediction comparison.
Figure 6. 25CrMo4 extrusion force prediction comparison.
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Figure 7. AISI52100 extrusion force prediction comparison.
Figure 7. AISI52100 extrusion force prediction comparison.
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Figure 8. AISI3310 extrusion force prediction comparison.
Figure 8. AISI3310 extrusion force prediction comparison.
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Figure 9. AISI316 die wear prediction comparison.
Figure 9. AISI316 die wear prediction comparison.
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Figure 10. H13 die wear prediction comparison.
Figure 10. H13 die wear prediction comparison.
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Figure 11. 25CrMo4 die wear prediction comparison.
Figure 11. 25CrMo4 die wear prediction comparison.
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Figure 12. AISI52100 die wear prediction comparison.
Figure 12. AISI52100 die wear prediction comparison.
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Figure 13. AISI3310 die wear prediction comparison .
Figure 13. AISI3310 die wear prediction comparison .
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Table 1. Physical and mechanical properties of the titanium alloy Ti6Al4V and magnesium alloy AZ31B [23,24].
Table 1. Physical and mechanical properties of the titanium alloy Ti6Al4V and magnesium alloy AZ31B [23,24].
Property Ti6Al4V AZ31B
Density (g/cm3) 4.46 1.74
Tensile strength (MPa) 895 260
Yield strength (MPa) 828 200
Elastic modulus (GPa) 110 44.80
Poisson’s ratio 0.31 0.35
Table 2. Table 2. Chemical composition of titanium alloy Ti6Al4V [23].
Table 2. Table 2. Chemical composition of titanium alloy Ti6Al4V [23].
Ti (wt.%) Al (wt.%) V (wt.%) Fe (wt.%) C (wt.%) O (wt.%) N (wt.%) H (wt.%)
Bal. 5.5–6.5 3.5–4.5 0.25 0.08 0.13 0.040 0.012
Table 3. Chemical composition of magnesium alloy AZ31B [24].
Table 3. Chemical composition of magnesium alloy AZ31B [24].
Mg (wt.%) Al (wt.%) Zn (wt.%) Mn (wt.%) Si (wt.%) Cu (wt.%) Ca (wt.%) Fe (wt.%) Ni (wt.%)
97 2.5–3.5 0.6–1.4 0.20 0.1 0.05 0.04 0.005 0.005
Table 4. Table 4. Chemical composition of die steels [25,26,27,28,29].
Table 4. Table 4. Chemical composition of die steels [25,26,27,28,29].
Material C (wt.%) Mn (wt.%) Si (wt.%) Cr (wt.%) Mo (wt.%) Ni (wt.%)
AISI316 0.08 2 0.75 16–18 2–3 10–14
H13 0.32–0.45 0.2–0. 5 0.8–1.20 4.75–5.50 1.10–1.75 0.30 max
25CrMo4 0.22–0.29 0.60–0.90 0.10–0.40 0.90–1.20 0.15–0.30 -
AISI52100 0.1 0.45 0.26 1.51 0.06 3.39
AISI3310 0.99 0.39 0.16 1.4 - 1.4
Table 5. Physical and mechanical properties of die steels [30].
Table 5. Physical and mechanical properties of die steels [30].
Property AISI316 H13 25CrMo4 AISI52100 AISI3310
Density (g/cm3) 8.03 7.78 7.85 7.83 7.81
Tensile strength (MPa) 550 1990 670 992 1866
Yield strength (MPa) 240 1650 435 579 1800
Elastic modulus (GPa) 210 210 205 200 210
Poisson’s ratio 0.3 0.3 0.3 0.3 0.3
Table 6. Process parameters ranking for extrusion force.
Table 6. Process parameters ranking for extrusion force.
Material Parameters
AISI316 Extrusion ratio, friction, ram speed, core diameter, billet height, die semi-angle and temperature.
H13 Friction, extrusion ratio, core diameter, billet height, die semi-angle, ram speed and temperature.
25CrMo4 Friction, ram speed, billet height, core diameter, die semi-angle, temperature and extrusion ratio.
AISI52100 Friction, core diameter, die semi-angle, extrusion ratio, billet height, ram speed and temperature.
AISI3310 Ram speed, core diameter, friction, extrusion ratio, die semi-angle, billet height and temperature.
Table 7. Determination factor (R2) for extrusion force linear regression model.
Table 7. Determination factor (R2) for extrusion force linear regression model.
Material R2
AISI316 0.91461
H13 0.92245
25CrMo4 0.70708
AISI52100 0.86922
AISI3310 0.91966
Table 8. Die materials ranking as function of the lowest extrusion force produced.
Table 8. Die materials ranking as function of the lowest extrusion force produced.
Material Ranking
AISI316 2
H13 2
25CrMo4 5
AISI52100 3
AISI3310 1
Table 9. Process parameters ranking for die wear.
Table 9. Process parameters ranking for die wear.
Material Parameters
AISI316 Friction, ram speed and temperature.
H13 Friction, ram speed and temperature.
25CrMo4 Temperature, friction and ram speed.
AISI52100 Friction, temperature and ram speed.
AISI3310 Temperature, ram speed and friction.
Table 10. Determination factor (R2) for extrusion force linear regression model.
Table 10. Determination factor (R2) for extrusion force linear regression model.
Material R2
AISI316 0.75695
H13 0.74873
25CrMo4 0.63223
AISI52100 0.80571
AISI3310 0.65881
Table 11. Die materials ranking as function of the minimum wear in the die produced.
Table 11. Die materials ranking as function of the minimum wear in the die produced.
Material Ranking
AISI316 2
H13 3
25CrMo4 N/A
AISI52100 N/A
AISI3310 1
Table 12. Die materials ranking as function of the minimum extrusion force and the minimum die wear.
Table 12. Die materials ranking as function of the minimum extrusion force and the minimum die wear.
Material Ranking
AISI316 2
H13 3
25CrMo4 5
AISI52100 4
AISI3310 1
Table 13. Table 13. Entropy method weights.
Table 13. Table 13. Entropy method weights.
W.
1(%)		 W.
2(%)		 W.
3(%)		 W.
4(%)		 W.
5(%)		 W.
6(%)		 W.
7(%)		 W.
8(%)		 W.
9(%)		 W.
10(%)		 W.
11(%)		 W.
12(%)		 W.
13(%)		 W.
14(%)		 W.
15(%)		 W.
16(%)		 W.
17(%)		 W.
18(%)
0.40 0.22 0.62 0 0.03 0.02 2.66 0.10 0.11 6.16 3.88 7.41 37.56 4.30 13.69 3.56 3.61 15.68
Table 14. Die materials VIKOR ranking.
Table 14. Die materials VIKOR ranking.
Material Ranking
AISI316 2
H13 3
25CrMo4 4
AISI52100 5
AISI3310 1
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