Data-Driven Design and Optimisation of Eutectic-Assisted Alumina Ceramics Using RSM and Symbolic Regression

Abstract

Lowering the processing temperature of alumina while retaining mechanical performance requires control of additive chemistry and powder processing. This study examines CaO–SiO₂–Al₂O₃-assisted alumina through a 27-condition dataset, response surface methodology (RSM) and symbolic regression using GPTIPS 2.0. Sintering temperature, coarse-alumina mass fraction, lubricant content and additive content are the input variables; flexural strength and density are performance responses, while shrinkage is treated as a dimensional response. The highest tabulated strength is 173.52 MPa at 1550 °C, with a density of 3.610 g/cm³. GPTIPS evaluation-subset variation explained is 89.55%, 87.39% and 81.24% for strength, density and shrinkage, respectively. These statistics describe model selection rather than independent predictive validation. A comparison with published reduced-cubic RSM models gives strength and density R² values of 0.9825 and 0.9934, whereas full-quadratic models yield lower adjusted R² values. The results demonstrate the influence of model specification on fitted agreement and distinguish the maximum tabulated response from a constrained literature optimum. The combined analysis provides a basis for assessing processing–property relationships while recognising the dimensional and service-performance trade-offs associated with liquid-phase-assisted sintering.

1. Introduction

Alumina is widely used in refractory, abrasive, electronic and biomedical applications because of its hardness, chemical stability and electrical insulation [1]. Optical applications also include carbon-doped alumina dosimeters [2]. Its thermal and mechanical performance depends on purity, porosity, grain structure and service conditions; thermal-shock resistance should therefore not be assumed to be uniformly high.

Alumina components are typically fabricated through the compaction or injection of alumina powders into moulds with the desired geometry, followed by sintering of the resulting green bodies at elevated temperatures. Consequently, significant attention has been directed in recent research towards reducing the sintering temperature without causing substantial degradation in the resulting ceramic properties [3,4,5]. The addition of low-melting-point additives, such as eutectic phases capable of forming intergranular liquid layers, has been shown to facilitate the sintering and bonding of ceramic particles at reduced temperatures [3]. These liquid-phase layers are able to wet the ceramic surfaces and infiltrate the interparticle spaces, leading to particle rearrangement that promotes ceramic densification. Furthermore, the surface tension associated with the liquid phase contributes to enhanced densification and assists in the elimination of pores within the ceramic matrix [6].

Liquid-phase-forming additions assist alumina densification, with the outcome depending on additive chemistry, quantity and processing conditions [3,4,5]. The CaO–Al₂O₃–SiO₂ phase equilibria provide a basis for selecting compositions with low liquid-formation temperatures [7].

In addition to the type and amount of the liquid phase, several other parameters—such as the particle size and size distribution of alumina, the type and concentration of the lubricant agent, the sintering temperature, and the soaking time—have also been shown to influence the final properties of sintered components. The conventional “one factor at a time” approach, which is often employed for optimising these variables, has been found to be both time-consuming and labour-intensive, as it requires a large number of experimental trials. Moreover, this method fails to account for the potential interactions among the influencing factors, rendering it inadequate for effective optimisation [8,9]. Therefore, the application of statistical methods for experimental design has been deemed necessary to reduce the number of required trials and, consequently, to minimise both time and cost. Among these methods, RSM has been widely employed. RSM is a statistical technique capable of modelling the relationship between multiple factors at varying levels and the corresponding responses, typically by fitting a polynomial equation that characterises these interactions [8].

Assuming that all factors are measurable, the response surface may be represented by the following general equation:

y = f(x₁͵x₂͵x₃͵…)

(1)

In this expression, y denotes the response, while x₁ to x_n represent the process variables or input factors influencing the response [10]. In essence, RSM comprises three main stages: (1) the design and execution of experiments, (2) the development of the RSM model using regression analysis, and (3) the optimisation of the response [11].

In recent years, considerable attention has been directed towards the application of RSM by researchers. For instance, RSM was employed by Li et al. [8] to optimise the relative density and bending strength of sintered CaO-partially stabilised zirconia (CaO-PSZ), using soaking time, sintering temperature, and heating rate as process variables. Similarly, Sktani et al. [9] applied this method to model and optimise the toughness and hardness of zirconia-toughened alumina (ZTA) ceramics, considering chemical composition, sintering time, and temperature as input factors. In addition, numerous studies have been conducted on the modelling and optimisation of sintering processes for materials such as Si₃N₄–SiC composites [12], titanium powder [10], high-speed steels (HSS) [13], and zirconia-toughened alumina (ZTA) composites [14].

Keramat and Hashemi [15] previously investigated the optimisation of liquid-phase sintering of alumina ceramics using central composite design (CCD) and response surface methodology (RSM). Building upon their published experimental dataset, the present study extends the analytical framework through genetic programming-based symbolic regression (GPTIPS 2.0) [16], incorporating sintering shrinkage as an additional response variable and comparing alternative empirical modelling approaches. The objective is to further investigate nonlinear relationships between processing parameters and ceramic properties, providing complementary insights beyond the previously reported RSM-based optimisation.

Recent work illustrates the range of available processing routes. Garanin et al. [17] combined intensive milling and a CuO–TiO₂–Nb₂O₅ additive to obtain ZTA at 1050 °C. Tang et al. [18] investigated Dy₂O₃–Al₂O₃–SiO₂ glass infiltration into dense alumina, while Stosz et al. [19] combined ternary additives with spark plasma sintering. These studies show why additive chemistry must be considered together with particle size, pressure and thermal history when comparing performance.

2. Materials and Methods

2.1. Synthesis of the CaO–SiO₂–Al₂O₃ Eutectic Phase

The synthesis was carried out through precise stoichiometric mixing and thermal processing. In this manner, to obtain the eutectic composition within the CaO–SiO₂–Al₂O₃ system, high-purity starting materials were employed, including calcium carbonate (CaCO₃, ~99% purity, ~600 nm), silicon dioxide (SiO₂, ~99% purity, ~7.8 μm), and kaolin (Al₂Si₂O₅(OH)₄, ~99% purity, ~1.8 μm). The powders were accurately weighed in stoichiometric proportions corresponding to a eutectic point with a melting temperature of approximately 1270 °C, as shown in Table 1. Initial mixing was performed via planetary ball milling in distilled water for 6 h using zirconia media to avoid contamination. The resulting slurry was dried at 110 °C for 12 h, ground to a fine powder, and calcined at 950 °C for 4 h in air to ensure complete decomposition of carbonate components and to initiate solid-state reactions. A subsequent high-temperature treatment at 1350 °C for 2 h was conducted to promote the formation of the eutectic phases anorthite, gehlenite, and pseudo-wollastonite [7].

Table 1. Nominal formulation of the ternary sintering additive.

Additive System	CaO (wt%)	Al2O3 (wt%)	SiO2 (wt%)	Basis for Composition Selection
CaO–Al2O3–SiO2	38	20	42	Composition investigated by Keramat and Hashemi [15]

Table 2 describes four additive heat treatments: 1200 and 1250 °C, each held for 4 or 8 h. The heat treatments at 1200 and 1250 °C were conducted below the nominal eutectic melting temperature of approximately 1270 °C, promoting solid-state reactions and the formation of the desired crystalline phases. Subsequent sintering of the alumina composites at higher temperatures facilitated liquid-phase formation, contributing to particle rearrangement and densification. The reported processing also includes attrition milling with zirconia media (ball-to-material mass ratio approximately 15:1, 800 rpm, 1 h), compaction at 10 MPa into 18 mm × 3 mm bodies, and pulverisation through a 120-mesh screen before XRD. XRD was conducted using a Bruker D8-ADVANCE with Cu Kα radiation at 40 kV and 40 mA.

Table 2. Sample designations and heat-treatment conditions investigated during preparation of the CaO–SiO₂–Al₂O₃ additive.

Treatment Temperature (°C)	Sample Designation: 4 h Holding Time	Sample Designation: 8 h Holding Time
1200	G1	G2
1250	G3	G4

Differential scanning calorimetry (DSC) was used to characterise the thermal behaviour of the G1–G4 additive preparations. The resulting thermograms were examined to assess differences in thermal response associated with the additive heat-treatment conditions. The DSC results are presented and discussed in Section 3.2.

2.2. Composite Preparation and Mechanical Testing

The fixed-composition additive comparison and the variable-factor RSM matrix are described separately. In the reported preliminary comparison, alumina powder (~99.5% purity; average particle size ~11.1 μm) was mixed with each G1–G4 additive at 93.5:6.5 by mass, followed by calcium stearate addition (reported as 1 wt%). The mixtures were ball milled before compaction. G0 denotes the comparison mixture prepared directly from CaCO₃, SiO₂ and kaolin without presynthesis of the additive; it is not an additive-free alumina control.

For the preliminary comparison, the reported green specimens were disks of 18 mm diameter and 3 mm thickness, pressed at 40 MPa and sintered at 1400 °C for 4 h in air. The RSM experimental design, presented in Table 3, investigated a wider sintering temperature range of 1400–1600 °C, together with variations in coarse-alumina fraction, lubricant content and eutectic additive content, to evaluate their effects on the mechanical and physical properties of the sintered ceramics. Ring-on-ring testing used a Santam universal testing machine at a crosshead speed of 1 mm/min [20]. The test configuration and concentric loading geometry are illustrated in Figure 1. For the preliminary comparison, at least ten specimens were tested for each composition to assess the mechanical performance of the sintered ceramics. The RSM analysis was conducted using the experimental response values presented in Table 3. Ring-on-ring loading measures specimen equibiaxial flexural strength, not an isolated interfacial bond strength. ASTM C1499-19 (2024) [21] provides the relevant test framework. The circular-disk expression below is also given by Keramat and Hashemi [15] and is included to make the load-to-stress conversion explicit:

$${\sigma }_f={\displaystyle \frac{3F}{2\pi h^2}}\left[\left(1-\nu \right){\displaystyle \frac{D_S^2 - D_L^2}{2D^2}}+\left(1+\nu \right)\ln \left({\displaystyle \frac{D_S}{D_L}}\right)\right]$$

(2)

Here F is the fracture load, h and D are the measured fired-specimen thickness and diameter, DS is the support-ring diameter, DL is the loading-ring diameter, and ν is Poisson’s ratio. Loads in N and dimensions in mm give stress in MPa. The expression assumes an appropriate concentric-ring geometry and plate response; contact friction, alignment, warpage and large deflection require assessment.

Table 3. Dataset organised by central composite design point type for the present modelling analysis. Coded inputs z₁–z₄ represent sintering temperature, coarse-alumina mass fraction, lubricant content and additive content, respectively; conversions to physical units are defined in Section 2.3. Processing inputs, flexural strength and density data are taken from Keramat and Hashemi [15].

Design Group	Source Run	z1	z2	z3	z4	Flexural Strength (MPa)	Density (g/cm3)	Shrinkage (%)
Factorial	1	+1	+1	−1	−1	113.63	3.319	28.34
Factorial	4	+1	+1	+1	−1	138.04	3.471	29.61
Factorial	7	+1	−1	+1	+1	173.52	3.610	29.04
Factorial	8	−1	−1	+1	+1	114.55	3.412	28.66
Factorial	14	+1	+1	+1	+1	132.11	3.459	27.48
Factorial	15	+1	+1	−1	+1	148.07	3.522	32.37
Factorial	16	−1	+1	+1	−1	130.54	3.346	29.68
Factorial	17	−1	−1	−1	−1	162.00	3.528	29.12
Factorial	18	−1	+1	−1	−1	141.75	3.476	36.41
Factorial	19	−1	+1	−1	+1	146.61	3.504	29.90
Factorial	20	+1	−1	−1	+1	113.33	3.289	23.43
Factorial	22	+1	−1	+1	−1	171.03	3.571	30.95
Factorial	23	+1	−1	−1	−1	113.87	3.346	30.16
Factorial	24	−1	−1	−1	+1	123.03	3.378	34.89
Factorial	25	−1	−1	+1	−1	105.00	3.225	29.08
Factorial	26	−1	+1	+1	+1	125.17	3.411	29.28
Axial	2	0	+2	0	0	98.74	3.197	20.91
Axial	3	0	0	0	−2	143.38	3.494	28.92
Axial	5	0	−2	0	0	132.13	3.448	28.45
Axial	9	0	0	+2	0	142.78	3.471	29.24
Axial	12	0	0	0	+2	150.60	3.521	30.13
Axial	13	−2	0	0	0	159.05	3.512	30.17
Axial	21	+2	0	0	0	111.32	3.307	23.18
Axial	27	0	0	−2	0	131.03	3.417	28.77
Centre	6	0	0	0	0	145.53	3.471	27.70
Centre	10	0	0	0	0	161.56	3.525	30.27
Centre	11	0	0	0	0	146.03	3.499	28.67

Table 3. Dataset organised by central composite design point type for the present modelling analysis. Coded inputs z₁–z₄ represent sintering temperature, coarse-alumina mass fraction, lubricant content and additive content, respectively; conversions to physical units are defined in Section 2.3. Processing inputs, flexural strength and density data are taken from Keramat and Hashemi [15].

Design Group	Source Run	z1	z2	z3	z4	Flexural Strength (MPa)	Density (g/cm3)	Shrinkage (%)
Factorial	1	+1	+1	−1	−1	113.63	3.319	28.34
Factorial	4	+1	+1	+1	−1	138.04	3.471	29.61
Factorial	7	+1	−1	+1	+1	173.52	3.610	29.04
Factorial	8	−1	−1	+1	+1	114.55	3.412	28.66
Factorial	14	+1	+1	+1	+1	132.11	3.459	27.48
Factorial	15	+1	+1	−1	+1	148.07	3.522	32.37
Factorial	16	−1	+1	+1	−1	130.54	3.346	29.68
Factorial	17	−1	−1	−1	−1	162.00	3.528	29.12
Factorial	18	−1	+1	−1	−1	141.75	3.476	36.41
Factorial	19	−1	+1	−1	+1	146.61	3.504	29.90
Factorial	20	+1	−1	−1	+1	113.33	3.289	23.43
Factorial	22	+1	−1	+1	−1	171.03	3.571	30.95
Factorial	23	+1	−1	−1	−1	113.87	3.346	30.16
Factorial	24	−1	−1	−1	+1	123.03	3.378	34.89
Factorial	25	−1	−1	+1	−1	105.00	3.225	29.08
Factorial	26	−1	+1	+1	+1	125.17	3.411	29.28
Axial	2	0	+2	0	0	98.74	3.197	20.91
Axial	3	0	0	0	−2	143.38	3.494	28.92
Axial	5	0	−2	0	0	132.13	3.448	28.45
Axial	9	0	0	+2	0	142.78	3.471	29.24
Axial	12	0	0	0	+2	150.60	3.521	30.13
Axial	13	−2	0	0	0	159.05	3.512	30.17
Axial	21	+2	0	0	0	111.32	3.307	23.18
Axial	27	0	0	−2	0	131.03	3.417	28.77
Centre	6	0	0	0	0	145.53	3.471	27.70
Centre	10	0	0	0	0	161.56	3.525	30.27
Centre	11	0	0	0	0	146.03	3.499	28.67

Figure 1. Ring-on-ring equibiaxial flexural testing: (a) exploded assembly; (b) diametral loading section; and (c) concentric contact geometry.

Figure 1. Ring-on-ring equibiaxial flexural testing: (a) exploded assembly; (b) diametral loading section; and (c) concentric contact geometry.

2.3. Method

The present study employs a four-factor central composite design (CCD) dataset comprising 27 experimental conditions to investigate processing–property relationships in eutectic-assisted alumina ceramics. The processing parameters, flexural strength and density values are derived from the experimental dataset published by Keramat and Hashemi [15]. These data are reanalysed using full-quadratic least-squares models and genetic programming-based symbolic regression (GPTIPS 2.0), while the previously published reduced-cubic RSM models provide a comparative benchmark. Sintering shrinkage is incorporated as an additional response to extend the evaluation of ceramic processing behaviour.

Symbolic regression using GPTIPS 2.0 [16] was used as a complementary empirical modelling approach. Agreement between RSM and GPTIPS fitted to the same measurements does not independently validate either the measurements or the proposed sintering mechanisms.

The physical input variables were:

• x1: Sintering temperature (°C).
• x2: Coarse-alumina mass fraction in the alumina powder blend (%).
• x3: Lubricant content (wt%).
• x4: Eutectic phase content (wt%).

Table 3 presents the processing inputs as dimensionless coded variables: z₁ = (T − 1500)/50, z₂ = (C − 50)/25, z₃ = (L − 1)/0.5 and z₄ = (E − 6.5)/1, where T is sintering temperature in °C, C is coarse-alumina mass fraction in mass%, and L and E are lubricant and additive contents in wt%, respectively. The same coding is used for the quadratic analysis in Appendix A and the literature-model reconstruction in Appendix B. Responses are retained in physical units. Rows are grouped as factorial, axial and centre points; this presentation order does not alter the source run identifiers or the input–response pairings used in the analyses.

The dataset comprising 27 samples was randomly divided into a training set (70%) and a test set (30%). Each response of fracture strength, density, and shrinkage was modelled separately. The GPTIPS configuration included the following:

• Function set: {+, −, ×, square, sqrt, exp}.
• Population size: 220 individuals.
• Number of generations: 120.
• Maximum tree depth: 4.
• Maximum number of genes: 5.
• Selection method: Tournament (size = 4).
• Fitness function: Root Mean Squared Error (RMSE).

The best evolved models were selected based on their predictive performance on the test set and their interpretability. Parity plots were generated to compare the predicted and actual values. The relative importance of genes and the structure of symbolic expressions were analysed to gain physical insights into the governing mechanisms. In this way, the Pseudo code shown in Algorithm 1 explains the procedure.

Algorithm 1. Explanation of the procedure employed to symbolically model the variation in fracture strength, density, and sintering shrinkage with Sintering_Temperature, Alumina_Particle_Size, Lubricant_Content, Glass_Phase_Content.

BEGIN

1. LOAD dataset

INPUT variables:      X = [Sintering_Temperature, Alumina_Particle_Size, Lubricant_Content, Glass_Phase_Content]    OUTPUT variables:      Y1 = Fracture_Strength      Y2 = Density      Y3 = Sintering_Shrinkage

2. INITIALIZE GPTIPS configuration

SET function_set = {+, −, ×, square, sqrt, exp}    SET population_size = 220    SET generations = 120    SET max_tree_depth = 4    SET max_genes = 5    SET selection_method = Tournament    SET tournament_size = 4    SET fitness_function = RMSE

3. FOR each target_variable in {Y1, Y2, Y3} DO

a. CREATE a GPTIPS model     SET gmodel.input = X     SET gmodel.target = target_variable

b. CONFIGURE gmodel parameters:     gmodel.funcSet = function_set     gmodel.popSize = population_size     gmodel.genCount = generations     gmodel.maxDepth = max_tree_depth     gmodel.maxGenes = max_genes     gmodel.selection = ‘tournament’     gmodel.tournamentSize = tournament_size     gmodel.fitnessFunction = ‘RMSE’

c. RUN symbolic regression using GPTIPS     gmodel = GPTIPS_RUN(gmodel)

d. EXTRACT best evolved model:     best_model = gmodel.bestIndividual

e. OUTPUT best_model expression and performance metrics

END FOR

4. DISPLAY symbolic models for all three outputs:    - Fracture Strength model    - Density model    - Sintering Shrinkage model

END

3. Results and Discussion

3.1. XRD

Figure 2 shows the XRD patterns associated with the G1–G4 heat treatments. The principal phase assignments are gehlenite, anorthite and pseudowollastonite. The residual silica assignment is expressed as α-quartz (SiO₂), distinguishing the crystalline polymorph from the generic oxide composition. The patterns provide qualitative phase information; phase fractions and glass content are not quantified by this comparison.

Figure 2. XRD patterns of the CaO–SiO₂–Al₂O₃ additive after heat treatment: (a) G1, 1200 °C for 4 h; (b) G2, 1200 °C for 8 h; (c) G3, 1250 °C for 4 h; and (d) G4, 1250 °C for 8 h. The open-diamond symbol denotes α-quartz (SiO₂).

3.2. Thermal Behaviour and Liquid-Phase Formation

Figure 3 compares the DSC thermograms of G1–G4. The traces exhibit a downward excursion followed by an upward peak, with exothermic heat flow directed upwards. The downward feature is less pronounced for G1 than for G2–G4. These differences indicate that the additive preparation conditions affect the subsequent thermal response. Peak shape alone does not establish phase identity, reaction completion or a quantitative ranking of additive homogeneity.

Figure 3. DSC thermograms of the G1–G4 additive preparations defined in Table 2. Heat flow is expressed in arbitrary units; exothermic heat flow is directed upwards.

Solid-state formation of the additive and liquid formation during composite sintering are separate stages. The 1200 and 1250 °C treatments lie below the nominal eutectic temperature of approximately 1270 °C [15], whereas the 1400–1600 °C composite-sintering interval lies above it. Thus, the lower-temperature preparation treatments are consistent with solid-state reaction, followed by liquid-phase-assisted densification during composite sintering.

The low-temperature liquid can promote wetting, particle rearrangement and pore elimination, while the final microstructure depends on composition, particle size and thermal history [3,4,5,6,7]. The nominal eutectic temperature is a literature-based composition benchmark and is not presented as a DSC onset or peak measured in this study.

3.3. Response Surface Analysis and Processing Benchmarks

The CCD comprises four processing factors at five coded levels (−2, −1, 0, +1 and +2). The analysis extends the previously published dataset [15] by examining the response trends using full-quadratic RSM and symbolic regression, with the published reduced-cubic models serving as a benchmark. The original strength and density measurements are retained to enable direct comparison between modelling approaches, while sintering shrinkage is considered as an additional response for evaluating dimensional changes during ceramic processing.

The powder-blend variable is C = 100 mc/(mc + mf), where mc and mf are the masses of coarse (mean size 11.1 μm) and fine (mean size 4.2 μm) alumina, respectively. Thus C = 25% means 25 mass% coarse and 75 mass% fine alumina within the alumina fraction, excluding lubricant and additive. C = 0% and 100% denote fully fine and fully coarse alumina. This variable is a blend mass fraction, not a particle diameter or a percentile of a particle-size distribution.

The design comprises 27 design points, consistent with the CCD count in Equation (3):

n_e = 2^k + 2k + l,

(3)

Here k = 4 is the number of factors and l = 3 is the number of centre-point runs. Table 3 groups the 16 factorial, eight axial and three centre-point runs separately. The centre points retain source run identifiers 6, 10 and 11.

Ordinary least squares were used to fit a full-quadratic model containing four linear terms, four squared terms and six two-factor interactions. Appendix A gives the coefficients and conditional ANOVA for all three responses. The R² values are 0.612, 0.594 and 0.408 for strength, density and shrinkage, respectively; adjusted R² values are 0.160, 0.121 and −0.283, with overall model p-values of 0.302, 0.350 and 0.827. The negative adjusted R² for shrinkage indicates that the quadratic basis offers little explanatory benefit after accounting for its parameter count. These statistics apply to the specified quadratic models.

For comparison, the 23-coefficient reduced-cubic basis specified in [15] was reconstructed from the published rounded values. Appendix B reports its retained terms, coefficients and model-level ANOVA. Strength and density R² values are 0.9825 and 0.9934, with adjusted R² values of 0.8864 and 0.9571. Figure 4 and Figure 5 illustrate the reconstructed literature strength and density surfaces and the quadratic shrinkage surface. Each panel varies two coded inputs between −1 and +1 while holding the other two at their centre values. The reduced-cubic fits have four residual degrees of freedom; their high fitted agreement is not an independent validation result.

Figure 4. Calculated strength surfaces from the reconstructed literature reduced-cubic model, using the coefficients in Appendix B. Panels (a–f) show factor pairs (T,C), (T,L), (T,E), (C,L), (C,E) and (L,E), respectively. The remaining factors are held at their centre values.

Figure 5. Calculated response surfaces for density (a–f), obtained from the reconstructed literature reduced-cubic model using Appendix B coefficients, and sintering shrinkage (g–l), obtained from the quadratic model using Appendix A coefficients: (a,g) sintering temperature and coarse-alumina mass fraction; (b,h) sintering temperature and lubricant content; (c,i) sintering temperature and additive content; (d,j) coarse-alumina mass fraction and lubricant content; (e,k) coarse-alumina mass fraction and additive content; and (f,l) lubricant content and additive content. In each panel, the remaining two factors are held at their centre values.

The maximum tabulated strength is 173.52 MPa at 1550 °C, 25% coarse alumina, 1.5 wt% lubricant and 7.5 wt% additive (source run 7 in Table 3); density and shrinkage are 3.610 g/cm³ and 29.04%. This tabulated maximum differs from the constrained optimum reported in [15], which is 1533 °C at the same remaining factor settings. The source predicts 172.49 MPa and 3.566 g/cm³ and reports confirmation measurements of 163.15 MPa and 3.767 g/cm³. These extracted literature values provide a processing benchmark. Appendix B summarises the source objective settings and distinguishes prediction from confirmation.

Comparisons with other processing routes are given in Table 4. Lower furnace temperature alone is not a complete measure of energy efficiency; milling, dwell time, pressure, additive preparation and final properties also matter. Different matrices, flaw populations and test geometries preclude a direct ranking of strength values across studies.

Table 4. Contextual benchmarks. Values are reported for different systems and test conditions and are not directly interchangeable.

System/Source	Processing	Reported Outcome	Comparison Limit
This Table 3, run 7	1550 °C; 7.5 wt% additive	173.52 MPa; 3.610 g/cm3	Tabulated maximum; same strength/density values as [15]; ROR specimen response
CuO–TiO2–Bi2O3/[22]	1200 °C; 90 min	97.63% relative density	Strength not stated in the consulted abstract; different chemistry
ZTA + CTN/[17]	1050 °C; intensive milling	235 MPa biaxial strength; 5.54% porosity	ZrO2 reinforcement and finer powders
Alumina + DAS infiltration/[18]	Glass distributed at boundaries at 1550 °C	525 ± 25 MPa flexural strength	Infiltration route; different glass and test conditions
Alumina + CTN/[19]	SPS: 750 °C, 60 MPa; pressureless: 1050 °C	Approximately 100% and 97% relative density, respectively	Pressure-assisted route is not a like-for-like furnace comparison

Residual grain-boundary glass can promote densification but may also permit viscous deformation and grain-boundary sliding at elevated temperature, alter dielectric loss, and change thermal stresses. These are composition-dependent risks, not measured outcomes of the present dataset. Glass-containing alumina can also show improved damage recovery under particular conditions [18]. No service-temperature limit can be assigned here without glass-transition/softening data, creep tests, dielectric measurements over frequency and temperature, and thermal-shock testing. Post-sintering crystallisation annealing is a possible future treatment, but must be evaluated for porosity, residual stress and mechanical degradation rather than presumed beneficial.

3.4. Symbolic Regression-Based Modelling of Ceramic Properties

GPTIPS 2.0 was used to explore nonlinear relationships between the four processing factors and the three responses. The resulting training and evaluation plots characterise the empirical model fit and the variation explained during model selection.

The 70:30 partition separates fitting and evaluation observations within the 27-row dataset. Because the evaluation subset also contributes to model selection, its reported performance is interpreted as an internal evaluation measure. The small dataset and repeated centre conditions limit the strength of conclusions about generalisation.

The symbolic-regression results are presented through the response-specific performance plots in Figure 6, Figure 7 and Figure 8. The plots use transformed response coordinates. Their numerical axes are therefore not interpreted directly as MPa, g/cm³ or percentage shrinkage, and no physical-unit RMSE is inferred from them.

Figure 6. GPTIPS strength outputs: response traces for (a) training and (b) evaluation observations; predicted-versus-observed values for (c) training and (d) evaluation observations; and (e) weighted gene contributions and (f) reported gene p-values. The plot labels “test” refer to the model-selection evaluation subset. Response coordinates are on the plotted transformed scale.

Figure 7. GPTIPS density outputs: response traces for (a) training and (b) evaluation observations; predicted-versus-observed values for (c) training and (d) evaluation observations; and (e) weighted gene contributions and (f) reported gene p-values. The plot labels “test” refer to the model-selection evaluation subset. Response coordinates are on the plotted transformed scale.

Figure 8. GPTIPS shrinkage outputs: response traces for (a) training and (b) evaluation observations; predicted-versus-observed values for (c) training and (d) evaluation observations; and (e) weighted gene contributions and (f) reported gene p-values. The plot labels “test” refer to the model-selection evaluation subset. Response coordinates are on the plotted transformed scale.

Figure 6, Figure 7 and Figure 8 report training/evaluation variation explained of 97.697/89.5498% for strength, 95.1576/87.3852% for density and 94.169/81.2396% for shrinkage. The evaluation measure is highest for strength and lowest for shrinkage. All three evaluation values are lower than their training values, consistent with reduced fitted agreement outside the fitting subset. These percentages describe the displayed GPTIPS outputs; they are distinct from the RSM goodness-of-fit statistics in Appendix A and Appendix B.

Shrinkage is treated as a process response related to dimensional change. Greater shrinkage is not necessarily desirable because excessive or nonuniform contraction may impair dimensional tolerance and shape retention. Future multi-response optimisation should constrain shrinkage using a specified dimensional requirement while targeting adequate strength and density.

Symbolic regression provides an empirical description of the observed response patterns. Model fit alone does not identify the physical mechanism controlling densification or fracture.

The RSM statistics describe fits to the complete tabulated dataset, whereas the GPTIPS percentages describe training and model-selection subsets. These evaluation settings differ, so the numerical values do not establish superiority of one modelling method over the other.

3.5. Microstructural Characterisation

3.5.1. EDX Elemental Distribution

Figure 9 presents the original EDX maps and associated BSE image for specimen S2. The Al, Si and Ca channels show spatially heterogeneous signals across the imaged region. The Al signal is concentrated mainly towards the lower part of the field, whereas Si and Ca signals extend across a broader area. These maps provide qualitative information about spatial distribution; they do not establish a calibrated concentration gradient, diffusion distance or reaction-layer composition.

Figure 9. Original EDX elemental maps and associated BSE image of the cross-sectional region of specimen S2. The displayed channels are Al Kα, Si Kα, Ca Kα, Mo Lα, Ni Kα and Ag Lα. Channel labels are retained as recorded; the maps are qualitative and are not quantitative concentration profiles. The BSE image includes a 20 μm scale bar.

The EDX maps also contain signals labelled Mo, Ni and Ag, which are not constituents of the nominal CaO–SiO₂–Al₂O₃ ceramic system. The Ag signal may be associated with conductive coating or silver-based mounting materials used during specimen preparation. Furthermore, spectral interference, background contributions or EDX peak misidentification may contribute to the additional elemental signals. As their precise origin cannot be confirmed from the available mapping results, these signals are not attributed to the ceramic composition or interfacial reaction products. The maps are therefore interpreted qualitatively and are not considered conclusive evidence of elemental interdiffusion or reaction-layer formation.

3.5.2. BSE Microstructural Observations

Figure 10 shows BSE cross-sections of specimen S2 at nominal magnifications of 500× and 1000×. Both images contain regions of contrasting grey level, including an irregular bright band and a heterogeneous lower region with dark features. The higher-magnification view resolves additional local variations in the band and surrounding material. The morphology is spatially nonuniform, so the images are not described as demonstrating a uniformly dense or defect-free interface.

Figure 10. Original BSE cross-sectional images of specimen S2 at nominal magnifications of (a) 500× and (b) 1000×. Scale bars are (a) 100 μm and (b) 50 μm. Regions of differing contrast are described morphologically without assigning phase identity from brightness alone.

The observed contrast is reported descriptively. Brightness alone is insufficient to identify alumina, a glassy phase or a reaction product, and the dark features are not assigned exclusively to porosity. The images therefore complement the specimen-level flexural results as qualitative structural observations; they do not measure interfacial bond strength or establish a diffusion-controlled consolidation mechanism.

4. Conclusions

This study applies symbolic regression and comparative response surface modelling to a 27-condition dataset for eutectic-assisted alumina ceramics, examining the relationships between processing parameters, flexural strength, density and sintering shrinkage.

The 27-condition analysis relates temperature, coarse-alumina fraction, lubricant content and additive content to strength, density and shrinkage. The highest tabulated strength is 173.52 MPa at 1550 °C, 25% coarse alumina, 1.5 wt% lubricant and 7.5 wt% additive. Shrinkage is treated as a dimensional response rather than a quantity to maximise.

Model specification strongly affects fitted agreement. Reconstruction of the published reduced-cubic strength and density models yields R² values of 0.9825 and 0.9934, whereas the full-quadratic models give substantially lower adjusted R². The 1533 °C processing condition is a constrained literature optimisation benchmark, distinct from the temperature of the maximum tabulated strength.

GPTIPS evaluation-subset variation explained is 89.55% for strength, 87.39% for density and 81.24% for shrinkage. These internal model-selection statistics support an exploratory description of the response patterns. They do not establish independent predictive validation or a causal microstructural mechanism.

Liquid-phase-assisted processing involves trade-offs between densification and the behaviour of residual grain-boundary phases. High-temperature creep, dielectric response and thermal-shock behaviour are relevant directions for further study, together with controlled post-sintering crystallisation. The present analysis does not assign a numerical service-temperature limit.