#
Applicability of Extreme Vertices Design in the Compositional Optimization of 3D-Printed Lightweight High-Entropy-Alloy/B_{4}C/ZrO_{2}/Titanium Trihybrid Aero-Composite

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{50}Cu

_{10}Sn

_{5}Mg

_{20}Zn

_{10}Ti

_{5}lightweight high-entropy alloy (LHEA), B4C, and ZrO

_{2}for the fabrication of trihybrid titanium composites via direct laser deposition. A mixture design was involved in the experimental design, and experimental data were modeled and optimized to achieve the optimal performance of the trihybrid composite. The ANOVA, response surface plots, and ternary maps analyses of the experimental results revealed that various combinations of reinforcement particles displayed a variety of response trends. Moreover, the analysis showed that these reinforcements significantly contributed to the magnitudes and trends of the responses. The generated models were competent for predicting response, and the best formulation consisted of 8.4% LHEA, 1.2% B4C, and 2.4% ZrO

_{2}.

## 1. Introduction

_{2}, TiO

_{2}, SiO

_{2}, Al

_{2}O

_{3}, TiB

_{2}, and BN are examples of reinforcements that have been included into titanium matrices [6,7], resulting in enhanced performance. Ceramic particles have high operating temperatures and moduli; hence, they are employed in metal composites to increase thermal stability [8,9]. In addition, their use in metal composites has enhanced the composites’ hardness, wear resistance, and corrosion resistance.

_{50}Cu

_{10}Sn

_{5}Mg

_{20}Zn

_{10}Ti

_{5}) possessing a density of 3.74 g/cm

^{3}was engaged as a supplemental reinforcement to B4C and ZrO

_{2}ceramic particle reinforcement in Ti6Al4V alloy in the fabrication of a trihybrid composite. Owing to its excellent strength-to-weight ratio, Ti6Al4V is widely employed in the aerospace industry for airframes, as well as engine and turbine components [28]. In an effort to increase its thermal stability and improve the high-temperature performance, ceramic particles B4C and ZrO

_{2}are considered reinforcements based on their high-temperature stability. Al

_{50}Cu

_{10}Sn

_{5}Mg

_{20}Zn

_{10}Ti

_{5}is a ductile LHEA particle; hence, its incorporation into the Ti6Al4V matrix was devised to counteract the brittleness of the ceramic particles.

_{4}C, and ZrO

_{2}percentage, were used in our investigation. In order to conduct a comprehensive factorial experimental investigation, 81 experimental runs were required, which is laborious and cost intensive. For the objective of reducing experimental costs and the associated labor, the mixture design approach was engaged in the design of experiments, as was the case in previous research [30,31,32]. Earlier works evaluated addressed the manufacture and design of HEA-reinforced metal composites without clear optimization processes and prediction models for the property responses, which is the major aim of this report.

_{2}for producing an optimal trihybrid titanium composite. This work hereafter focuses critically on response surface analysis, development and validation of mathematical models for future response prediction, and multi-objective optimization of obtained experimental results.

## 2. Experimental Program

#### 2.1. Mixture Design of Experiment (MoE)

_{2}dose is expressed by factor C. The value range for each reinforcement is 0 to 12 wt.%. For the mixed design experiment, Design expert 13 software was used, and Table 1 displays the experimental runs for the design. As seen, the total of the components in each of the ten formulations was 12 wt.%. Therefore, this suggests that the mixture design for the trihybrid titanium composite is envisaged with a particle count of 12, according to [36]. As initially stated, this report intends to establish the optimal combination of the three components that will yield the best mechanical property balance for Al

_{50}Cu

_{10}Sn

_{5}Mg

_{20}Zn

_{10}Ti

_{5}/B4C/ZrO

_{2}/Ti6Al4V trihybrid composites. Mechanical parameters tested include tensile strength (TS), compressive strength (CS), elastic modulus (EM), hardness (HD), elastic strain (Es), and compressive strain (CS) (Cs). Following Equation (1), we generated the quadratic regression models for each response.

#### 2.2. Materials and Trihybrid Composite Fabrication

_{50}Cu

_{10}Sn

_{5}Mg

_{20}Zn

_{10}Ti

_{5}LHEA, B4C, and ZrO

_{2}powders, with particle sizes ranging from 7–18 μm. In the manufacturing procedure, direct laser deposition (DLD) was used, and the titanium-based plate was heated to 150 °C. The printing laser (ROFIN DL013S) with a wavelength of 800–940 nm was engaged, and the argon carrying gas flow rate was maintained at 0.1 L per second. In accordance with Table 1, the powders were pre-combined in the hopper, and a constant powder flow rate of 5 g/min was maintained throughout the printing operation. Based on the results of a prior experiment [33], the laser power was 500 W, and the speed was maintained at 800 mm/s. To accomplish a rise in height with each successive layer, a single layer with a length of 120 mm and an average thickness of 100 μm was deposited with an accuracy of ±0.005 mm for the average height of the layer. After the procedure, printed specimens of 120 × 120 × 100 mm were cut to various shapes (by laser machining) for the property tests. For the basis of comparison, pure Ti6Al4V with no reinforcement was prepared at the same condition.

#### 2.3. Property Examination

^{−3}s

^{−1}using a load of 100 KN, in line with the ASTM E 8M-21 [37] procedure. Three samples were subjected to each test representing each formulation, and the mean value for tensile strength and elastic modulus was recorded.

^{−3}s

^{−1}strain, engaging a 200 KN load. The average result of three samples (as regards compressive strength and strain) representing each formulation was collected.

## 3. Results and Discussion

#### 3.1. Analysis of Experimental Results

_{4}C, and 0% ZrO

_{2}exhibited no noticeable increase in tensile strength. Runs 2 through 10 exhibited a significant increase in tensile strength compared with the unreinforced alloy, indicating that the introduction of various particle formulations improved tensile strength compared with the unreinforced alloy. Run 7’s formulation of 12 wt.% LHEA, 0 wt.% B4C, and 0 wt.% ZrO

_{2}exhibited the highest tensile strength. The ductile LHEA at 12 wt.% (1512 MPa) is very favorable to tensile strength.

_{2}are optimal for compressive strength response. The elastic modulus of every mixture combination was larger than that of the unreinforced alloy. Consequently, all formulations used between runs 1 and 10 increased the elastic modulus. During Run 7, the highest value for elastic modulus was determined (160 GPa). Similar to tensile strength, 12 wt.% LHEA, 0 wt.% B4C, and 0 wt.% ZrO

_{2}exhibited the greatest increase in elastic modulus.

_{2}and 0 wt.% LHEA, 6 wt.% B4C, and 6 wt.% ZrO

_{2}are detrimental to hardness. Meanwhile, runs 3 through 10 acquired higher hardness than the pure alloy. The formulation containing 12% LHEA, 0% B4C, and 0% ZrO

_{2}exhibited a maximum hardness of 3.87 GPa. The pure alloy (ref) had the greatest elastic and compressive strain values (Figure 2e,f). Addition of the particles in various formulations reduced the elastic and compressive ductility, resulting in lower strain values as compared with the pure alloy. Conclusively, Figure 2 revealed disparities in the magnitude of the responses for different mixture combinations, necessitating response and formulation optimization to obtain balanced performance.

#### 3.2. Analysis of Variance

_{4}C outperformed other inputs. In terms of compressive strength and compressive strain, ZrO

_{2}also beat the other input materials.

#### 3.3. Predictive Models

#### 3.3.1. Predicted Values vs. Actual (Experimental Values)

#### 3.3.2. Mathematical Models

#### 3.4. Probability Plot

#### 3.5. Predicted vs. Experimental Value Plots

^{2}for each response parameter is greater than 0.90, indicating that the mathematical models are statistically adequate for predicting the values of response parameters under random conditions.

#### 3.6. Response Surface Plots

_{2}) are displayed on the horizontal x, y, and z axes of the surface plots, while the responses are exhibited on the vertical axis. Due to the examination of six response characteristics, six surface plots were generated (Figure 6): one for each response parameter.

_{2}increased the tensile strength (Figure 6a) to a maximum of 1402 MPa at (12, 5, 6) wt.% coordinate. The mixture of 0 to 12 wt.% LHEA, 5–12 wt.% B4C, and 6–12 wt.% ZrO

_{2}enhanced tensile strength, resulting in a minimum value of 1160 MPa, which corresponds to (12, 12, 0) wt.%. LHEA exhibited a positively convex interaction profile, while B4C and ZrO

_{2}exhibited an inverted “U”-shaped parabolic pattern with inflexion points at 5 and 6 wt.%, respectively. The graph revealed that the tensile strength was reliant on the interaction profile of the three independent variables.

_{2}is synergistic to compressive strength, resulting in a maximum value of 1160 MPa at a coordinate combination of 12 wt.% LHEA, 5 wt.%B4C, and 6 wt.% ZrO

_{2}for each input. The same proportions of B4C and ZrO

_{2}coupled with 0–4% LHEA led to a decrease in compressive strength. A 4–12 wt.% combination of the input factors resulted in a decrease in strength. B4C and ZrO

_{2}exhibited an inverted “U”-shaped parabolic line of fit. Since there is no independent interaction, it is evident that the compressive strength is contingent on the pattern of interaction between the variables.

_{2}enhanced elastic modulus, whereas the same range of LHEA and B4C with 6–12 wt.% ZrO

_{2}decreased elastic modulus. ZrO

_{2}showed an inverted “U”-shaped parabolic outline with an inflexion point at 6 wt.%, corresponding to 160 GPa. The maximum elastic modulus was achieved at position 12% LHEA, 12% B4C, and 6% ZrO

_{2}, which corresponds to 173 GPa, while the lowest was 142.5 GPa at coordinate 0, 0, 0. Consequently, this demonstrates an increase in elastic modulus and a dependency of the response on the interaction pattern between factors.

_{2}is antagonistic to hardness, whereas the blend of 5–12 wt.% LHEA, 0–4 wt.% B4C, and 6–12 wt.% ZrO

_{2}resulted in an improvement of hardness. The maximum hardness was observed to be 12 wt.% LHEA, 4 wt.% B4C, and 12 wt.% ZrO

_{2}, yielding a value of 3.93 GPa, even though a minimum value of 2.5 GPa was reached at a coordinate combination of 5 wt.% LHEA, 6 wt.% B4C, and 6 wt.%. LHEA and ZrO

_{2}exhibited a “U”-shaped parabolic profile, while B4C depicted an inverted “U”-shaped parabolic interaction pattern. The analysis is indicative of the dependence of the hardness of the trihybrid composite on the behavioral pattern of the independent variables.

_{2}showed a progressive decrease in elastic strain with a minimum strain at coordinates (8, 8, 12), equaling 0.81 × 10

^{–3}mm/mm. On the other hand, 8–12 wt.% LHEA, 8–12 wt.% B4C, and 0–6 wt.% ZrO

_{2}ensue an increase in elastic strain with a maximum value of 1.23 × 10

^{−3}mm/mm at coordinates (0, 0, 6). While LHEA and ZrO

_{2}delayed the “U”-shaped parabolic profile, B4C exhibited an inverted “U”-shaped pattern. Elastic strain is established to depend on the interaction patterns of the input variables.

_{2}admixture results in a rise in compressive strain of which a 6–12 wt.% LHEA, 0–12 wt.% B4C, and 6–12 wt.% ZrO

_{2}mixture provoked a reduction in compressive strain. Maximum compressive strain is realized at 6 wt.% LHEA, 12 wt.% B4C, and 6 wt.% ZrO

_{2}(9.42 × 10

^{−2}mm/mm), and minimum compressive strain at 0 wt.% LHEA, 0 wt.% B4C, and 12 wt.% ZrO

_{2}. While LHEA and ZrO

_{2}displayed an inverted “U”-shaped parabolic profile, B4C portrayed a positive convex profile. Similar to other response parameters, the magnitude of compressive strain is hinged on the behavioral pattern of the input variables.

#### 3.7. Ternary Maps for Responses

#### 3.8. Multiobjective Optimization and Model Validation

_{2}). The software (design expert 13) predicts the optimal parameters to be 1431.87 MPa, 1109.35 MPa, 157.806 GPa, 3.53074 GPa, 1.07422 × 10

^{−2}mm/mm, and 8.61695 × 10

^{−3}mm/mm for tensile strength, compressive strength, elastic modulus, hardness, elastic strain, and compressive strain, respectively, with a maximum desirability of 0.733.

_{2}, respectively. The samples were evaluated for the assessed properties, and the mean results for each property response were recorded. The experimental values for the respective responses are 1403.9 MPa, 1138.7 MPa, 163 GPa, 3.48 GPa, 1.10 × 10

^{−2}mm/mm, and 8.38 × 10

^{−3}mm/mm. The variations between actual and predicted values for each property are 1.95, 2.65, 3.29, 1.44, 2.40, and 2.75% (Figure 9). The difference between the anticipated values and experimental values is 5%; hence, the mathematical models for the responses are further validated and deemed suitable for future prediction of the property parameters evaluated in this research.

## 4. Conclusions

_{50}Cu

_{10}Sn

_{5}Mg

_{20}Zn

_{10}Ti

_{5}, B4C, and ZrO

_{2}fillers was conducted utilizing a mixed design experiment. The following is arrived at:

- The analysis of variance revealed that the contributions of the inputs to the investigated response properties are substantial. The relevance of the cross-interaction between the input components varied for each response.
- Mathematical models were built for the attributes, and their mathematical significance in forecasting the responses was confirmed.
- The response surface plot revealed various interaction patterns for the input variables and their interactions, indicating that the trend of the responses depended on the interactive patterns between the variables.
- The ternary plots revealed varying regions in achieving varying outputs at varying parameter combinations.
- The predicted optimization formulation was 8.43695 wt.% for A (LHEA), 1.19982 wt.% for B (B4C), and 2.36332 wt.% for C (ZrO
_{2}), yielding tensile strength, compressive strength, elastic strain, and compressive strain values of 1431.87 MPa, 1109.35 MPa, 157.806 GPa, 3.53074 GPa, 1.07422 × 10^{−2}mm/mm, and 8.61695 mm/mm.

_{4}C, and ZrO

_{2}, based on the highest desirability of 0.733 obtained.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Variation in experimental values at varying formulation of the reinforcement with regards to (

**a**) tensile strength, (

**b**) compressive strength, (

**c**) elastic modulus, (

**d**) hardness, (

**e**) elastic strain, and (

**f**) compressive strain.

**Figure 3.**Percentage contribution of the input powders on response properties of the trihybrid titanium composite.

**Figure 6.**Surface plots for response parameters: (

**a**) tensile strength, (

**b**) compressive strength, (

**c**) elastic modulus, (

**d**) hardness, (

**e**) elastic strain, and (

**f**) compressive strain.

**Figure 7.**Ternary plots for response parameters: (

**a**) tensile strength, (

**b**) compressive strength, (

**c**) elastic modulus, (

**d**) hardness, (

**e**) elastic strain, and (

**f**) compressive strain.

Experimental Runs | Formulation | Total | ||
---|---|---|---|---|

A (wt.%) | B (wt.%) | C (wt.%) | ||

0 (control) | 0 | 0 | 0 | 0 |

1 | 0 | 12 | 0 | 12 |

2 | 6 | 0 | 6 | 12 |

3 | 6 | 6 | 0 | 12 |

4 | 4 | 4 | 4 | 12 |

5 | 0 | 0 | 12 | 12 |

6 | 2 | 8 | 2 | 12 |

7 | 12 | 0 | 0 | 12 |

8 | 8 | 2 | 2 | 12 |

9 | 0 | 6 | 6 | 12 |

10 | 2 | 2 | 8 | 12 |

Tensile Strength | Compressive Strength | Elastic Modulus | Hardness | Elastic Strain | Compressive Strain | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Source | p-Value | %C | p-Value | %C | p-Value | %C | p-Value | %C | p-Value | %C | p-Value | %C |

A + B + C | <0.0001. | 36.5 | 0.0131 | 24.3 | 0.0002 | 48.9 | 0.0031 | 19.8 | <0.0001 | 26.9 | 0.0001 | 32.7 |

AB | 0.0004 | 30.2 | 0.0170 | 12.6 | 0.0081 | 24.6 | 0.0038 | 18.6 | <0.0001 | 28.2 | 0.0006 | 27.78 |

AC | 0.2498 | 15.1 | 0.0002 | 27.9 | 0.0745 | 2.3 | 0.0002 | 33.5 | 0.0069 | 25.5 | 0.0001 | 35.9 |

BC | 0.0013 | 18.2 | <0.0001 | 35.2 | 0.0081 | 24.2 | 0.0014 | 28.1 | 0.0202 | 19.4 | 0.8930 | 3.6 |

Error | 0.9873 | 0.64 | 1.1329 | 0.61 | 0.8863 | 1.06 | 0.7886 | 1.21 | 1.2284 | 0.44 | 1.1377 | 0.61 |

Model | 0.0002 | 99.36 | 0.0002 | 99.39 | 0.0005 | 98.94 | 0.0006 | 98.79 | <0.0001 | 99.56 | 0.0002 | 99.39 |

Experimental Runs and Values | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

TS | PV | 1005.97 | 1297.56 | 1414.28 | 1374.75 | 1141.24 | 1277.56 | 1506.70 | 1445.37 | 1294.83 | 1284.74 |

EV | 1010.00 | 1287.00 | 1408.00 | 1402.00 | 1141.00 | 1272.00 | 1512.00 | 1436.00 | 1283.00 | 1291.00 | |

% error | −0.40 | +0.81 | +0.44 | −1.98 | +0.02 | +0.44 | −0.35 | +0.65 | +0.91 | −0.57 | |

CS | PV | 1050.05 | 1125.81 | 1072.99 | 1158.25 | 1022.87 | 1132.99 | 1039.50 | 1109.63 | 1185.36 | 1141.54 |

EV | 1052.00 | 1120.00 | 1071.00 | 1160.00 | 1021.00 | 1131.00 | 1040.00 | 1112.00 | 1181.00 | 1151.00 | |

% error | −0.19 | +0.52 | +0.19 | −0.15 | +0.18 | +0.18 | −0.05 | −0.21 | +0.37 | −0.83 | |

EM | PV | 139.88 | 152.01 | 153.82 | 151.50 | 128.07 | 147.46 | 159.88 | 157.46 | 142.01 | 142.91 |

EV | 140.00 | 153.00 | 155.00 | 152.00 | 128.00 | 146.00 | 160.00 | 156.00 | 143.00 | 142.00 | |

% error | −0.09 | −0.65 | −0.77 | −0.33 | +0.05 | +0.99 | −0.08 | +0.93 | −0.70 | +0.64 | |

HD | PV | 3.01 | 3.17 | 3.75 | 3.56 | 3.69 | 3.51 | 3.88 | 3.61 | 3.75 | 3.55 |

EV | 3.01 | 3.16 | 3.74 | 3.50 | 3.69 | 3.53 | 3.87 | 3.66 | 3.75 | 3.57 | |

% error | 0.00 | +0.32 | +0.27 | +1.69 | 0.00 | −0.57 | +0.26 | −1.39 | 0.00 | −0.56 | |

Es | PV | 0.9873 | 1.2100 | 0.8065 | 0.9842 | 1.2000 | 0.9096 | 1.1200 | 1.0200 | 1.0200 | 1.1200 |

EV | 0.9900 | 1.2100 | 0.8100 | 1.0000 | 1.2000 | 0.9000 | 1.1200 | 1.0000 | 1.0200 | 1.1200 | |

% error | −0.27 | 0.00 | −0.43 | −1.61 | 0.00 | +1.06 | 0.00 | +1.96 | 0.00 | 0.00 | |

Cs | PV | 8.33 | 8.60 | 8.74 | 8.55 | 7.50 | 8.47 | 8.18 | 8.64 | 7.91 | 8.14 |

EV | 8.34 | 8.60 | 8.75 | 8.50 | 7.50 | 8.44 | 8.16 | 8.70 | 7.94 | 8.14 | |

% error | −0.12 | 0.00 | −0.11 | +0.58 | 0.00 | +0.35 | +0.24 | −0.69 | −0.38 | 0.00 |

Parameter | Desirable Region | Range of Values for Independent Variable Combination | Predicted Range of Values | Predicted Desirable Values | ||
---|---|---|---|---|---|---|

LHEA (wt.%) | B_{4}C (wt.%) | ZrO_{2} (wt.%) | ||||

Tensile strength | A | 9.07–12.0 | 1.02–2.93 | 0.00–0.39 | 1500–1600 MPa | 1517.11 MPa |

Compressive strength | B | 0.00–5.11 | 2.50–9.22 | 2.77–8.63 | 1150–1200 MPa | 1185.57 MPa |

Elastic modulus | C | 0.00–5.24 | 0.00–8.03 | 0.00–6.69 | 150–160 GPa | 159.83 GPa |

Hardness | D | 0.05–12 | 0.00–5.15 | 0.00–9.83 | 3.80–4.00 GPa | 3.88 GPa |

Elastic strain | E | 0.00–7.04 | 0.00–0.38 | 0.00–12.00 | 1.20–1.30 mm/mm | 1.225 mm/mm |

Compressive strain | F | 0.00–9.76 | 0.00–9.76 | 0.00–6.00 | 8.60–8.80 mm/mm | 8.737 mm/mm |

Name | Goal | Lower Limit | Upper Limit |
---|---|---|---|

A: LHEA | is in range | 0 | 12 |

B: B_{4}C | is in range | 0 | 12 |

C: ZrO_{2} | is in range | 0 | 12 |

Tensile strength | maximize | 1010 | 1512 |

Compressive strength | maximize | 1021 | 1181 |

Elastic modulus | maximize | 128 | 160 |

Hardness | maximize | 3.01 | 3.87 |

Elastic strain | maximize | 0.81 | 1.21 |

Compressive strain | maximize | 7.5 | 8.75 |

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**MDPI and ACS Style**

Akinwande, A.A.; Moskovskikh, D.; Romanovskaia, E.; Balogun, O.A.; Kumar, J.P.; Romanovski, V.
Applicability of Extreme Vertices Design in the Compositional Optimization of 3D-Printed Lightweight High-Entropy-Alloy/B_{4}C/ZrO_{2}/Titanium Trihybrid Aero-Composite. *Math. Comput. Appl.* **2023**, *28*, 54.
https://doi.org/10.3390/mca28020054

**AMA Style**

Akinwande AA, Moskovskikh D, Romanovskaia E, Balogun OA, Kumar JP, Romanovski V.
Applicability of Extreme Vertices Design in the Compositional Optimization of 3D-Printed Lightweight High-Entropy-Alloy/B_{4}C/ZrO_{2}/Titanium Trihybrid Aero-Composite. *Mathematical and Computational Applications*. 2023; 28(2):54.
https://doi.org/10.3390/mca28020054

**Chicago/Turabian Style**

Akinwande, Abayomi Adewale, Dimitry Moskovskikh, Elena Romanovskaia, Oluwatosin Abiodun Balogun, J. Pradeep Kumar, and Valentin Romanovski.
2023. "Applicability of Extreme Vertices Design in the Compositional Optimization of 3D-Printed Lightweight High-Entropy-Alloy/B_{4}C/ZrO_{2}/Titanium Trihybrid Aero-Composite" *Mathematical and Computational Applications* 28, no. 2: 54.
https://doi.org/10.3390/mca28020054