Prediction of Physical and Mechanical Properties of Al₂O₃–TiB₂–TiC Composites Using Design of Mixture Experiments

Abstract

In this study, the design of mixture experiments was used to find empirical models that could predict, for a first approximation, the relative density, flexural strength, Vickers hardness and fracture toughness of sintered composites in order to identify further areas of research in the Al₂O₃-TiB₂-TiC ternary system. The composites were obtained by spark plasma sintering (SPS) of these mixtures at 1700 °C, 80 MPa and a dwell of 3 min. The obtained experimental results were analyzed in the statistical analysis software Minitab 17, and then, different regression models were obtained for each property. Based on the selected models, contour plots were made in the Al₂O₃–TiB₂–TiC simplex for a visual representation of the predicted results. By combining these plots, it was possible to obtain one common zone in the Al₂O₃–TiB₂–TiC simplex, which shows the following combination of physical and mechanical properties for sintered samples: relative densities, flexural strength, Vickers hardness, and fracture toughness of than 99%, 500 MPa, 18 GPa, and 7.0 МPa·m^1/2, respectively. For a first approximation in determining the further area of research, the obtained models describe well the behavior of the studied properties. The results of the analysis showed that the design of mixture experiments allows us to identify the most promising compositions in terms of mechanical properties without resorting to labor-intensive and financially expensive full-scale experiments. Our work shows that 10 different compositions were required for preliminary analysis.

1. Introduction

In the past few decades, ceramic materials such as alumina (Al₂O₃), titanium carbide (TiC), and titanium diboride (TiB₂) have been attracting more and more attention and have greater application as key components of ceramic tool materials, due to their excellent properties such as high hardness, strength, as well as high wear, corrosion, and heat resistances [1]. Nevertheless, the application of single phases of these ceramic materials is limited by their poor fracture toughness and variability in their mechanical strength.

Some scholars have reported that composites based on these phases have higher hardness, flexural strength, and fracture toughness compared to monolithic materials. The ternary composite materials based on Al₂O₃–TiB₂–TiC are commonly produced by self-propagating high temperature synthesis (SHS) [2], laser-assisted SHS [3], hot pressing (HP) [4,5,6], or spark plasma sintering (SPS) [7]. The results of these works indicate that mechanical properties of these ternary composites are superior to those of their single-phase materials, and even to those of the binary composites based on Al₂O₃–TiC [8]; Al₂O₃–TiB₂ [9]; TiB₂–TiC [10].

Al₂O₃–TiB₂–TiC-based composite materials may have a variety of characteristics, depending on which phase is the main component. For instance, high alumina content would lead to a composite material for corrosion applications that shows higher fracture toughness, superior mechanical strength, and better oxidation and impact resistance. On the other hand, composites based on TiC or TiB₂ could be used for tribological and electrical applications, due to their enhanced wear resistance and electrical conductivity [11]. Thus, composites based on the Al₂O₃–TiB₂–TiC system, due to their improved properties, are attractive for advanced applications as wear-resistant coatings [12,13,14], strengthening of metal–ceramic composites [11,15,16], ceramic cutting tools [4,5,6,17], and as self-lubricating ceramic tool material [18]. Thereby, the preliminary determination of the physical and mechanical properties of compositions within the ternary diagram Al₂O₃–TiB₂–TiC is necessary to establish a rational choice of composite composition depending on the intended application.

To prevent the change of the TiB₂ phase into titanium (TiO₂) and boron (B₂O₃) oxides due to prolonged high temperatures in the presence of aluminum oxide, as well as to achieve high density values, it is necessary to use a sintering technology that will ensure a high heating rate with simultaneous mechanical pressing. SPS technology can provide high heating rates (from 100 °C/min to 1000 °C/min) that allow the conventional long process of sintering materials to be transformed into a fast and short process, in which uncontrolled grain growth is reduced or eliminated [19,20,21,22,23]. Furthermore, the application of external mechanical pressure during heating permits high density values of sintered materials with better mechanical properties [24,25]. However, full-scale experiments are expensive and time-consuming, so alternative methods are needed.

The design of mixture experiments is a method which uses statistical concepts applied to mixture problems to find relations between chemical constituents as the factors and mixture properties as the responses in order to identify the behavior of each component in the mixture environment [26]. Once relationships are established, which is commonly completed through mathematical models, they can be used to describe, predict, or explain results, as well as to optimize compositions [27]. According to the design of mixture experiments, to obtain a linear mathematical model of a ternary system, only three experimental points are necessary; adding three other experimental points, it is possible to establish a second order polynomial, and a reduced cubic model can be calculated with one more experimental point [27]. This shows that the design of mixture experiments is a good tool to establish the relationships between factors and the responses using a minimum number of trials in the mixture environment.

This approach is sometimes used to predict the physical and mechanical properties of ceramic composites in the first approximation with the aim of further optimization of the composition depending on the required properties of the study. For example, de Mestral and Thevenot showed in their research [28,29,30] how, using the design of mixture experiments, they could predict the tensile strength, hardness, fracture toughness, and electrical conductivity of boride–carbide composites. In addition, other research groups used this method to predict the other properties of ceramic composites, such as water absorption, flexural strength, open porosity, linear firing shrinkage, weight loss, density, hardness, friction, and wear behavior [31,32,33,34].

The aim of this work was to establish mathematical models that could predict, for a first approximation, the relative density, flexural strength, Vickers hardness, and fracture toughness of sintered composites in order to identify further areas of research in the Al₂O₃–TiB₂–TiC ternary system. The established models could be used for the elaboration of contour plots in the Al₂O₃–TiB₂–TiC ternary diagram for a visual representation of the predicted results and for the rapid identification of zones or compositions with specific required properties.

2. Materials and Methods

2.1. Powder Mixture Preparation

In this study, three different commercial ceramic powders, Al₂O₃ (purity 99.9%), TiB₂ (purity 99.9%), and TiC (purity 99.5%), produced by “Plasmotherm” Ltd., Moscow, Russia, were used.

A {3,2} centroid simplex-lattice design (Figure 1), augmented with interior points, was generated in Minitab 17 in order to define the composition of the necessary mixtures that should be investigated in this work. The expression {q,m} specifies that this design can support a polynomial model of degree “m” using “q” components in the simplex-lattice [19]. Moreover, the number “m” is used to determine “m + 1” equal proportions of each component that vary from 0 to 1, and which are calculated as follows:

$$x_i=0,{\displaystyle \frac{1}{m}},{\displaystyle \frac{2}{m}},\dots ,1.$$

In this way, {3,2} indicates that it is possible to obtain a second-degree model, which can represent the response surface over the entire simplex region consisting of 3 components with the use of 3 equal proportions for each of them (0, 1/2, 1). Thus, the points of the {3,2} simplex-lattice are

$$(x_1,x_2,x_3)=\left(1,0,0\right),(1/2,1/2,0),(0,1,0),(0,1/2,1/2),(0,0,1),(1/2,0,1/2)$$

that correspond to points No. 1, 2, 3, 4, 5, 6 of this work (Figure 1), respectively. The notations $x_1$, $x_2$, $x_3$ represent the proportions of Al₂O₃, TiC, and TiB₂, respectively.

Furthermore, a central point (No. 7, Figure 1) and three testing points (No. 8–10, Figure 1) were included in the {3,2} simplex-lattice, with the aim to design an augmented {3,2} centroid simplex-lattice.

These 10 points in the ternary plot Al₂O₃–TiB₂–TiC show the coordinates of each mixture, and their coordinates indicate the amount-of-substance fraction in molar % (

Table 1. Coordinates of mixtures in the ternary plot Al₂O₃–TiB₂–TiC.

Point	Al2O3	TiC	TiB2	Mixture
No.		[mol %]		Name
1	100.00	0.00	0.00	ATB-1
2	50.00	50.00	0.00	ATB-3
3	0.00	100.00	0.00	ATB-5
4	0.00	50.00	50.00	ATB-7
5	0.00	0.00	100.00	ATB-9
6	50.00	0.00	50.00	ATB-11
7	33.33	33.33	33.33	ATB-22
Test points
8	67.00	16.50	16.50	ATB-13
9	16.50	67.00	16.50	ATB-16
10	16.50	16.50	67.00	ATB-19

).

For every mixture, the proportions of each raw powder materials, listed in Table 1, were added to a polyethylene jar. Al₂O₃ balls (∅3 mm) were then added to the jar with powders at a ball-to-powder weight ratio of 3:1. Isopropanol, as the liquid medium, was then also added to the jar at an isopropanol-to-powder weight ratio of 1:1. Next, the polyethylene jars were tightly sealed and placed in a ball mill to grind and mix the components of each powder mixture for 36 h of use. After this time, the obtained wet mixtures were dried in a vacuum drying oven for 12 h at 80 °C. Finally, the dried mixtures were crushed in an agate mortar and then sifted through a 63-micron sieve.

2.2. Spark Plasma Sintering

All powder mixtures were sintered in a Spark Plasma Sintering Machine H-HP D 25 SD from FCT Systeme GmbH (Rauenstein, Germany), and disc composites with a diameter of 40 mm and 4 mm in height were obtained. The sintering process started with the application of a pressure of 47 MPa that was maintained from room temperature up to 300 °C. After that, both pressure and temperature grew continuously up to 80 MPa and 1600 °C, respectively. Heating from room temperature up to 1600 °C was conducted by a heating rate of 100 °C/min. After reaching 1600 °C the heating rate was reduced to 25 °C/min to reach the sintering temperature of 1700 °C which was maintained for 3 min. When heating finished, samples were cooled naturally in the sintering chamber. Figure 2 shows the applied force and temperature schedules, as well as the punch displacement behavior as a function of time during the SPS process. From each powder mixture, three sintered samples were obtained, in which the measurements of relative density and flexural strength were carried out for statistical analysis.

2.3. Relative Densities Measurement

Archimedes’ method was used for measuring the sintered composite densities in distilled water. For each mixture, the relative density was calculated as the ratio of its measured density over the calculated theoretical density. Theoretical densities were calculated according to the rule of mixtures assuming densities of 3.94 g/cm³ for Al₂O₃, 4.85 g/cm³ for TiC, and 4.35 g/cm³ for TiB₂, which were measured by an AccuPyc II 1340 helium pycnometer (Micrometrics, Norcross, GA, USA).

2.4. Flexural Strength Testing

Flexural strength was determined following the standard [35] using a displacement rate of 0.5 mm/min in an ElectroPuls E10000 universal testing machine (Instron, High Wycombe, UK) by three-point bend tests, with a span of 20 mm. The samples for the tests were bars with a cross section of 2.0 mm × 1.5 mm and length of at least 25 mm. In this regard, the end surfaces of the sintered discs with a diameter of 40 mm and a height of 4 mm were initially flat grinded, in order to achieve three goals: removing the defective layer, achieving a disc thickness of 2 mm, and flat parallelism of the machined surfaces. In addition, due to the fact that the TiB₂ and TiC phases are electrically conductive, most of the sintered discs have sufficient electrical conductivity for wire electrical discharge machining (WEDM). Therefore, the electrically conductive discs were processed on a WEDM machine, and those discs that did not have sufficient electrical conductivity were processed on a precision cutting machine Accutom 50 (Struers GmbH, Ballerup, Denmark) using a diamond disc. Figure 3 shows the WEDM process of conductive specimens.

After WEDM of disks, the resulting bars were assembled and glued onto a metal substrate, and then the electrical discharge machining surfaces were flat grinded in order to remove defects on them, as well as to achieve plane parallelism and a size of 1.5 mm.

Figure 4 shows the sample reparation process of conductive specimens for the flat grinding. Figure 4a demonstrates the preparation and classification of obtained bars for gluing. Figure 4b shows the gluing process of bars to a metal substrate, while in Figure 4c, the flat grinding process of bars is exhibited.

After flat grinding, the bars were ground and polished before testing, and four edges on the tensile surface were chamfered to an angle of 45° in order to eliminate a failure initiated from the edge of the specimen. For each composition, 3 samples were tested, and the mean value of the test results was calculated.

2.5. Vickers Hardness Testing

The sintered samples were polished before the hardness measuring. The indentation method was implemented for measuring the Vickers hardness (Hv) of samples. The measuring process was carried out in a microhardness tester (Qness, Salzburg, Austria) with a standard diamond pyramid indenter, under a load of 98 N for 10 s. In each sample, 10 indentations were made, and then the arithmetic mean for each composition was calculated. The Equation used for the hardness calculation was

HV = 0.1891 P/d²,

where P is the set load (N), and d is the average length of two diagonals (mm).

2.6. Fracture Toughness Testing

Fracture toughness of the polished samples was measured by the microindentation method with an applied load of 98 N for 10 s. Fracture toughness was calculated using the formula given by Miranzo and Moya as was indicated in our previous work [1].

2.7. X-Ray Diffraction (XRD) Analysis

The determination of the phase composition of the sintered composites was carried out on an Empyrean X-ray diffractometer (PANalytical, Almelo, The Netherlands), in the following modes: Cu-Kα spectrum, wavelength 1.5405981 Å, voltage 60 kV, beam current 30 mA, in a 2θ range from 20° to 65°, and step size 0.05°.

3. Results

3.1. X-Ray Diffraction (XRD)

Figure 5 shows the XRD pattern of sintered composites from mixtures listed in Figure 1: ATB-1 (a), ATB-3 (b), ATB-5 (c), ATB-7 (d), ATB-9 (e), ATB-11 (f), ATB-13 (g), ATB-16 (h), ATB-19 (i), ATB-22 (j).

3.2. Relative Densities of Sintered Composites

Table 2. Relative densities of sintered composites.

Point	Mixture	ρth. *	Replication			Average
			1	2	3	Density
No.		[g/cm3]	ρrel, [%] *			ρrel, [%] *
1	ATB-1	3.950	97.53	96.69	98.36	97.53 ± 0.84
2	ATB-3	4.241	97.32	97.42	96.11	96.95 ± 0.73
3	ATB-5	4.850	94.52	93.52	93.79	93.94 ± 0.52
4	ATB-7	4.568	99.47	99.87	99.96	99.77 ± 0.26
5	ATB-9	4.350	99.35	98.20	98.70	98.75 ± 0.58
6	ATB-11	4.103	99.64	98.44	99.78	99.29 ± 0.74
7	ATB-22	4.334	98.89	97.95	98.31	98.38 ± 0.47
Test points
8	ATB-13	4.109	98.67	98.32	97.63	98.21 ± 0.53
9	ATB-16	4.498	98.86	99.61	99.96	99.48 ± 0.56
10	ATB-19	4.285	98.54	99.82	98.48	98.95 ± 0.76

*: ρ_th.—calculated theoretical density; ρ_rel—relative density.

shows the calculated theoretical densities for each powder mixture, as well as the measured relative densities of sintered composites.

3.3. Flexural Strength

The flexural strength test results are presented in

Table 3. Flexural strength of sintered composites.

Point	Mixture	Replication			Average
		1	2	3	Flexural Strength
No.		σf, Mpa *			σf, Mpa *
1	ATB-1	259.01	272.20	309.11	280.11 ± 25.97
2	ATB-3	460.05	507.42	409.23	458.90 ± 49.11
3	ATB-5	246.71	227.91	298.23	257.62 ± 36.41
4	ATB-7	532.86	561.88	639.03	577.92 ± 54.87
5	ATB-9	510.15	650.72	496.86	552.58 ± 85.25
6	ATB-11	399.75	447.12	410.28	419.05 ± 24.87
7	ATB-22	374.37	336.49	423.06	377.97 ± 43.40
Test points
8	ATB-13	351.83	413.64	474.68	413.38 ± 61.43
9	ATB-16	499.69	535.16	441.13	491.99 ± 47.49
10	ATB-19	601.10	670.64	637.39	636.38 ± 34.78

*: σf—measured flexural strength.

.

3.4. Vickers Hardness

Table 4. Vickers hardness of sintered composites.

Point	Mixture	Replication			Average
		1	2	3	Vickers Hardness
No.		HV, Gpa *			HV, Gpa *
1	ATB-1	15.2	14.0	12.5	13.9 ± 1.4
2	ATB-3	16.2	15.7	14.5	15.5 ± 0.9
3	ATB-5	14.2	14.8	12.5	13.8 ± 1.2
4	ATB-7	17.5	16.5	16.8	16.9 ± 0.5
5	ATB-9	15.9	16.9	17.5	16.8 ± 0.8
6	ATB-11	17.8	16.6	18.9	17.8 ± 1.2
7	ATB-22	21.3	18.9	21.1	20.4 ± 1.3
Test points
8	ATB-13	16.4	18.1	16.9	17.1 ± 0.9
9	ATB-16	17.2	19.9	19.4	18.8 ± 1.4
10	ATB-19	17.3	17.7	20.2	18.4 ± 1.6

*: HV—Vickers hardness.

shows the measured Vickers hardness of each sintered sample.

3.5. Fracture Toughness

The fracture toughness (K1c) test results are listed in

Table 5. Fracture toughness of sintered composites.

Point	Mixture	Replication			Average
		1	2	3	Fracture Toughness
No.		K1c, MPa·m1/2 *			K1c, MPa·m1/2 *
1	ATB-1	5.3	5.6	4.7	5.2 ± 0.5
2	ATB-3	6.7	6.4	7.0	6.7 ± 0.3
3	ATB-5	5.4	5.6	6.2	5.7 ± 0.4
4	ATB-7	5.7	5.8	6.1	5.9 ± 0.2
5	ATB-9	7.5	7.9	6.8	7.4 ± 0.6
6	ATB-11	7.7	6.7	6.4	6.9 ± 0.7
7	ATB-22	7.0	7.6	7.8	7.5 ± 0.4
Test points
8	ATB-13	9.1	8.8	7.8	8.6 ± 0.7
9	ATB-16	6.9	6.1	7.1	6.7 ± 0.5
10	ATB-19	7.8	8.2	8.1	8.0 ± 0.2

*: K1c—fracture toughness.

.

4. Discussion

The XRD patterns of sintered composites show that only Al₂O₃, TiB₂, and TiC phases were detected in the samples without the presence of any new impurities and phases.

The results of relative density, flexural strength, Vickers hardness, and fracture toughness were processed in Minitab 17, which is a statistical analysis software which can help in determining different regression models and establishing which of them is the most appropriate for the prediction of the given properties.

4.1. Relative Density Models

Table 6. Regression models obtained after ANOVA for relative density.

Regression Model	Property Equation
Linear	RDL = 98.06 X1 + 96.25 X2 + 100.06 X3
Quadratic	RDQ = 97.76 X1 + 94.47 X2 + 98.82 X3 + 4.83 X1X2 + 13.32 X2X3
Special cubic	RDSC = 97.43 X1 + 94.48 X2 + 98.49 X3+ 5.74 X1X2 + 3.88 X1X2 + 14.23 X2X3 − 14.4 X1X2X3

shows the regression models obtained after analysis of variance (ANOVA) of relative density. In this table, the abbreviation RD is related to relative density, while the subscripts L, Q, and SC are related to the type of model obtained: linear, quadratic, and special cubic, respectively.

In these models, the variables X1, X2, and X3 are coded values for Al₂O₃, TiC, and TiB₂, respectively. The coded values were calculated as the molar % of each component from Figure 1, divided by 100 (X1 = Al₂O₃ mol%/100; X2 = TiC mol%/100; X3 = TiB₂ mol%/100).

The estimated coefficients of the linear regression model (RD_L) were significant since ANOVA indicated that their p-values were lower than 0.05. However, its obtained R-squared (R²) and predicted R-squared (R²pred) values were 37.78% and 18.52%, respectively.

On the other hand, in the quadratic regression model (RD_Q), the quadratic term X1X3 was excluded from the equation, because its p-value was 0.234. For this model, the obtained R² and R²pred values were 72.95% and 63.86%, respectively. This indicates that RD_Q better fit the experimental data than RD_L.

The analysis of mixture design in terms of the special cubic model corroborated that the term X1X3 was not significant, because its p-value was 0.165 (higher than 0.05), and it should have been excluded from the model. Furthermore, this analysis showed that the cubic term X1X2X3 was also not significant (p-value = 0.430), and should also have been excluded from the model. As a result of the analysis, the obtained model RD_SC after exclusion of terms X1X3 and X1X2X3 took the form of a quadratic equation, coinciding with RD_Q. The R² and R²pred values of the reduced RD_SC were similar to these values for the RD_Q model.

Through a comparison between the model and experimental data, we have assessed the goodness of fit of the RD_Q model using residual plots. Figure 6a is a plot of standardized residuals against fitted values. These residuals represent the difference between the value of experimental responses of each composite and the estimated/fitted value of the density as predicted by the regression RD_Q model. If the residuals are roughly normally distributed, around 95% of them should lie between the cut-off values of −2.0 and 2.0, which are presented as horizontal lines. Values lying far above or below these boundaries are outliers. In Figure 6a, two residuals stand out from the basic pattern of residuals, and this suggests they are outliers. Moreover, Figure 6b shows the fitted line plot, which demonstrates a linear relationship between experimental responses and fitted values. According to the R-square value and the fitted line plot in Figure 6b, we can consider the model as significant.

Based on the obtained results, it is possible to conclude that, as a first approximation, the behavior of the relative density of sintered samples in the ternary plot Al₂O₃–TiB₂–TiC could be modeled by the quadratic equation RD_Q.

4.2. Flexural Strength Models

Similarly,

Table 7. Regression models obtained after ANOVA for flexural strength.

Regression Model	Property Equation
Linear	FSL = 326.70 X1 + 390.80 X2 + 622.30 X3
Quadratic	FSQ = 273.30 X1 + 270.30 X2 + 571.00 X3 + 623 X1X2 + 605 X2X3
Special cubic	FSSC = 282.30 X1 + 264.10 X2 + 580.00 X3 + 760.00 X1X2 + 742.00 X2X3 − 2226.00 X1X2X3

shows the regression models obtained after ANOVA in relation to flexural strength. In these equations, the abbreviation FS is related to flexural strength, while the subscripts L, Q, and SC are related to the type of model obtained: linear, quadratic, and special cubic, respectively.

The variables X1, X2, and X3 were calculated as indicated in Section 4.1.

In the linear regression model (FS_L), all the estimated coefficients were significant as their p-values were lower than 0.05. The obtained R² and R²pred values of this regression model were 47.42% and 32.72%, respectively.

In the quadratic regression model (FS_Q), the quadratic term X1X3 was excluded from the equation, because its p-value was 0.617. For this model, the obtained R² and R²pred values were 72.23% and 63.28%, respectively. This indicates that FS_Q better fit the experimental data than FS_L.

The analysis in terms of the special cubic model showed that the term X1X3 was not significant (p-value = 0.792), and it should have been excluded from the model. After exclusion of the term X1X3, the model FS_SC took the form shown in Table 7. The R² and R²pred values of this FS_SC model were 76.01% and 63.74%, respectively. These values indicated that FS_SC fit the experimental data better than FS_Q.

Figure 7a shows the plot of standardized residuals against fitted values obtained from the special cubic model FS_SC. In this plot, only two residuals are outliers. Moreover, Figure 7b shows the fitted line plot of the FS_Q model, which demonstrates a linear relationship between experimental responses and fitted values. According to the R-square value and the fitted line plot shown in Figure 7b, we can consider the model as significant.

Based on the obtained results, it is possible to conclude that, as a first approximation, the behavior of the flexural strength of sintered samples in the ternary plot Al₂O₃–TiB₂–TiC could be modeled by the cubic model FS_SC.

4.3. Vickers Hardness Models

Table 8. Regression models obtained after ANOVA for Vickers hardness.

Regression Model	Property Equation
Linear	HVL = 15.92 X1 + 16.17 X2 + 18.75 X3
Quadratic	HVQ = 13.55 X1 + 14.03 X2 + 16.37 X3+ 11.78 X1X2 + 13.92 X1X3 + 11.81 X2X3
Special cubic	HVSC = 13.72 X1 + 14.20 X2 + 16.54 X3 + 6.78 X1X2 + 8.92 X1X3 + 6.81 X2X3 + 81.00 X1X2X3

shows the models obtained in relation to Vickers hardness (HV), while the subscripts L, Q, SC, and FC are related to the type of model obtained: linear, quadratic, special cubic, and full cubic, respectively.

The calculations of variables X1, X2, and X3 was indicated in Section 4.1.

Since the p-values of the estimated coefficients in the linear regression model (HV_L) were lower than 0.05, it can be determined that they are significant. The obtained R² and R²pred values of HV_L were 15.09% and 0.00%, respectively. These values indicate that the linear model fit 6.83% with the experimental data, the model cannot predict new observations, and it cannot be considered for further research.

The HV_Q showed R² and R²pred values of 63.64% and 45.27%, respectively. This indicates that this model fits the experimental data much better than HV_L, but not sufficiently. The p-values of the estimated coefficients in this model were lower than 0.05.

The analysis of the experimental data in terms of the special cubic model (HV_SC) showed higher values of R² and R²pred were obtained (76.74% and 62.51%, respectively). This means that the HV_SC model fits the experimental data better than HV_Q.

The residual plot for the model HV_SC is shown in Figure 8a. This figure indicates that no residual is an outlier. The fitted line plot of the HV_SC model is demonstrated in Figure 8b, which shows a linear relationship between experimental responses and fitted values. According to the R-square value and the fitted line plot shown in Figure 8b, this model can be considered as significant.

Based on the obtained results, it is possible to conclude that the behavior of the Vickers hardness of sintered samples in the ternary plot Al₂O₃–TiB₂–TiC could be modeled, as a first approximation, by the cubic model HV_SC.

4.4. Fracture Toughness Models

Table 9. Regression models obtained after ANOVA for fracture toughness.

Regression Model	Property Equation
Linear	FTL = 6.73 X1 + 6.11 X2 + 7.74 X3
Quadratic	FTQ = 5.45 X1 + 5.44 X2 + 7.30 X3+ 7.52 X1X2 + 5.50 X1X3
Special cubic	FTSC = 5.53 X1 + 5.33 X2 + 7.19 X3 + 5.80 X1X2 + 3.78 X1X3 + 28.10 X1X2X3

shows the regression models obtained after ANOVA for fracture toughness, (FT), while the subscripts L, Q, SC, and FC are related to the type of model obtained: linear, quadratic, special cubic, and full cubic, respectively.

The variables X1, X2, and X3 can be calculated as indicated in Section 4.1.

All the estimated coefficients in the linear regression model (FS_L) were significant. The obtained R² and R²pred values of this regression model were 17.70% and 0.00%, respectively. These results indicate that this model fits the experimental data worst, and it cannot predict new observations. Thus, the FS_L model cannot be considered for further research.

The quadratic regression model (FT_Q) showed R² and R²pred values of 55.60% and 37.16%, respectively. This indicates that FT_Q fits the experimental data much better than FT_L, but not sufficiently. In this model, the quadratic term X2X3 was excluded from the equation because its p-value was 0.713.

The analysis in terms of the special cubic model showed that the term X2X3 was not significant (p-value = 0.146), and should have been excluded from the model. After exclusion of the term X2X3, the model FS_SC took the form shown in Table 9. The R² and R²pred values of this FT_SC model were 63.55% and 45.52%, respectively. These values indicated that FT_SC fit the experimental data better than FT_Q.

The comparison of the experimental data and fitted values from the FT_SC model for fracture toughness is presented in Figure 9a by the residual plot. From this figure, it can be seen that only two residuals are outliers. Moreover, Figure 9b shows a linear relationship between experimental responses and fitted values in the fitted line plot of the FT_SC model. According to the R-square value and the fitted line plot shown in Figure 9b, we can consider the model as significant.

Based on the obtained results, it is possible to conclude that the behavior of the fracture toughness of sintered samples in the ternary plot Al₂O₃–TiB₂–TiC could be modeled, as a first approximation, by the cubic model FT_SC.

4.5. Contour Plots

Based on the obtained equations RD_Q, FS_SC., HV_SC, and FT_SC, contour plots were made in the Al₂O₃-TiB₂-TiC simplex for a visual representation of the predicted results of the relative density, flexural strength, Vickers hardness, and fracture toughness of sintered samples (Figure 10).

Figure 10a shows the contour plot of relative density. From this figure, it can be noticed that the predicted zone with a density greater than 99% occupies an area which includes the points No. 4, 5, 7, and 10 (red numbers in the figure). According to this figure, the highest predicted values are found in the region between the points No. 4 and 5. Figure 10b shows the contour plot of flexural strength. The predicted zone with a flexural strength greater than 500 MPa occupies an area which includes the points No. 4, 5, and 10 and areas close to them. In this figure, it is possible to notice that values greater than 600 MPa are located in the region close to the points No. 4 and 5. Figure 10c shows the contour plot of Vickers hardness. Here, it can be noticed that predicted zone with a Vickers hardness greater than 18 GPa occupies an extended area, which includes the points No. 6, 7, 8, 9, and 10 and areas close to them. Moreover, the highest predicted values are found in the region close to point No. 7. Figure 10d shows the plot of the fracture toughness. The predicted zone with fracture toughness values greater than 7.00 МPa·m^1/2 occupies an extended area, which includes the points No. 5, 6, 7, 8, 9, and 10 and areas close to them. Here it is noticed that values greater than 8.00 МPa·m^1/2 are located in the region close to point No. 7.

By combining the plots in Figure 8, it is possible to obtain one common zone in the Al₂O₃-TiB₂-TiC simplex, which can predict the following combination of physical and mechanical properties for the sintered composites by SPS at 1700 °C, 80 MPa, and dwell of 3 min: relative densities, flexural strength, Vickers hardness, and fracture toughness greater than 99%, 500 MPa, 18 GPa, and 7.00 МPa·m^1/2, respectively (Figure 11).

The models RD_Q, FS_SC., HV_SC, and FT_SC were additionally tested at three control points (Figure 11 points No. 11–13), the coordinates of which are given in

Table 10. Coordinates of control points in the ternary plot Al₂O₃–TiB₂–TiC.

Point	Al2O3	TiC	TiB2	Mixture
No.		[mol %]		Name
11	11.1	40.3	48.6	2ATB-4
12	11.1	23.5	65.4	2ATB-5
13	16.7	46.0	37.3	2ATB-8

.

Table 11. Comparison of experimental predicted data of test points.

Point	ρrel [%]		95% CI	σf [MPa]		95% CI *	HV [GPa]		95% CI *	K1c [MPa·m1/2]		95% CI *
No.	exp. *	pred. *		exp. *	pred. *		exp. *	pred. *		exp. *	pred. *
11	99.57	99.77	(99.053; 100.491)	560	550.5	(498.2; 602.7)	19.03	19.16	(18.220; 20.108)	6.91	7.10	(6.549; 7.646)
12	99.49	99.85	(99.221; 100.479)	623	568.5	(523.6; 613.5)	18.93	18.93	(18.121; 19.741)	7.38	7.33	(6.858; 7.800)
13	98.77	99.30	(98.661; 99.933)	526	506.8	(453.2; 560.3)	18.85	19.56	(18.589; 20.530)	6.76	7.40	(6.838; 7.966)

*: exp.—experimental value; pred.—predicted value; CI—confidence interval.

shows the measured values of the relative density, flexural strength, Vickers hardness, and fracture toughness of sintered samples in the new three control points, as well as their predicted values, and confidence intervals.

From Table 11, it is clearly seen that experimental values fit well into the predicted intervals, except for the flexural strength in point No. 12. This can be related with the fact that the predicted R-squared value for the model FS_Q is 63.74%. The variation in predicted values for material relative density, flexural strength, and fracture toughness relative to experimental data shown in Table 11 can be explained by using the mean square pure error and residual error for each property and the chosen model. It is noticeable from

Table 12. Comparison of MSE for fitted and experimental values.

Mean Square	Relative Density	Flexural Strength	Vickers Hardness	Fracture Toughness
Pure error	0.3833	2443	1.333	0.2250
Residual error	0.9346	4589	1.475	0.4984

that the mean square residual errors of flexural strength and fracture toughness are approximately two times higher than the mean square pure error. Moreover, the mean square residual error of the relative density is approximately 2.5 times higher than the pure error. On the other hand, the error difference for Vickers hardness is smaller, and this explains the absence of a difference between the predicted and experimental data.

Despite these discrepancies in the experimental and fitted results, as a first approximation for determining the further direction of research, the selected models describe well the behavior of the studied properties of samples obtained by SPS at 1700 °C, 80 MPa, and a holding time of 3 min. Additionally, these results indicated that the design of mixture experiments is a simple and effective method for predicting, as a first approximation, physical and mechanical properties of ceramic composite materials with the use of only 10 different compositions for a composition of three components.

Moreover, in

Table 13. Theoretical and experimental density data of test points.

Point	ρth. *	ρ * [g/cm3]		95% CI *
No.	[g/cm3]	exp.*	pred.*
11	4.436	4.417	4.426	(4.394; 4.458)
12	4.369	4.347	4.362	(4.335; 4.390)
13	4.420	4.366	4.389	(4.361; 4.417)

*: ρ_th.—calculated theoretical density; ρ_—measured density; exp.—experimental value; pred.—predicted value; CI—confidence interval.

are shown the theoretical densities of sintered composites in the three control points, as well as their measured and predicted values, and their confidence intervals. From this table, it is clearly seen that experimental values fit well into the predicted intervals.

These results indicated that the design of mixture experiments is a simple and effective method for predicting, as a first approximation, the physical and mechanical properties of ceramic composite materials with the use of only 10 different compositions for a composition of three components.

5. Conclusions

In this study, the design of mixture experiments was used to find empirical models that could predict, for a first approximation, the relative density, flexural strength, Vickers hardness, and fracture toughness of sintered composites in order to identify further areas of research in the Al₂O₃-TiB₂-TiC ternary system.

The composites were obtained by spark plasma sintering at 1700 °C, 80 MPa, and dwell of 3 min. The measured properties of sintered composites were analyzed in Minitab 17, and then different regression models were established for each property. Among the found models, four of them (RD_Q, FS_SC, HV_SC, and FT_SC) were selected for the prediction of the studied properties.

In this work, the evaluation of the selected models was provided by the use of the residual plots and fitted line plots. This evaluation showed that the models can predict, as a first approximation, the behavior of the studied properties.

Based on the selected models, contour plots were made in the Al₂O₃-TiB₂-TiC simplex for a visual representation of the predicted results. By combining these plots, it was possible to obtain one common zone in the simplex, which shows the combination of relative densities, flexural strength, Vickers hardness, and fracture toughness greater than 99%, 500 MPa, 18 GPa, and 7.0 МPa·m^1/2, respectively.

Furthermore, the selected models were additionally tested at three control points in the established zone in the simplex. In these three points, the predicted values of relative density, flexural strength, and fracture toughness show a variation from the experimental data. It was demonstrated that this fact was related with the mean square errors of the models, which were higher than the mean square error of the experimental data.

Despite the fact that some fitted values were predicted with imperfect accuracy (for example, the fracture toughness), the selected empirical models allowed us to determine, for a first approximation, further areas of research. Moreover, the obtained results indicate that the design of mixture experiments is a simple and effective method for predicting the physical and mechanical properties of ceramic composite materials using a small number of different compositions.

The selected mathematical models work only for the prediction of properties of composites obtained by SPS within the specified sintering parameters and for the studied ceramic system.

In future work, the selected models will be refined in further extended studies conducted in the established area in the Al₂O₃-TiB₂-TiC simplex. A large number of mixtures will also be used and a more accurate model will be obtained. In addition, future work will include the study of microstructures to better understand mechanical characteristics.