Statistical Data-Driven Model for Hardness Prediction in Austempered Ductile Irons

Abstract

This research evaluates the effect of temperature and time austempering on microstructural characteristics and hardness of ductile iron, validating the results by means of a statistical method for hardness prediction. Ductile iron was subjected to austenitization at 950 °C for 120 min and then to austempering heat treatment in a salt bath at temperatures of 290, 320, 350 and 380 °C for 30, 60, 90 and 120 min. By increasing austempering temperature, a higher content of carbon-rich austenite was obtained, and the morphology of the thin acicular ferrite needles produced at 290 °C turned completely feathery at 350 and 380 °C. A thickening of acicular ferrite needles was also observed as austempering time increased. An inversely proportional behavior of hardness values was thus obtained, which was validated through data analysis, statistical tools and a regression model taking temperature and time austempering as input variables and hardness as the output variable, which achieved a correlation among variables of about 97%. The proposal of a mathematical model for the prediction of hardness in austempered ductile iron represents a numerical approximation which validates the experimental results at 95.20%.

1. Introduction

Austempered ductile irons (ADIs) are noticeable materials in the industrial market since the spheroidal shape of graphite provides high fatigue resistance compared to other graphitic morphologies, such as lamellar in gray cast iron, offering better continuity of the iron metal matrix [1]. Likewise, austempering heat treatment affords an excellent combination of properties, e.g., high toughness, good ductility and high wear resistance [2,3]. The successful production of high-performance components from ADI requires a solid understanding of microstructural changes caused by alterations in processing parameters, such as austempering heat treatment variables and chemical composition, as they considerably modify mechanical properties.

Currently, the industrial sector aims to satisfy economic and ecological initiatives of cost and environmental footprint reduction in manufacturing processes by creating materials with excellent mechanical properties and low density and proposing new materials to replace those commonly used [3]. ADIs are attractive as substitutes for steel and aluminum components due to low manufacturing costs and lower environmental impact as well as mechanical properties comparable to steel and superior to aluminum [3,4,5,6,7,8].

The outstanding mechanical properties of ADI derive from the unique ausferritic microstructure composed of graphite nodules (G), carbon-rich austenite (γ_hC) and acicular ferrite (α_ac). These phases are yielded through the application of a thermal cycle comprising:

• Ductile iron is heated to an austenitizing temperature (T_γ), between 850 and 1200 °C, for a sufficient time so as to homogenize the austenitic microstructure.
• Subsequently, it is rapidly cooled in a molten salt bath to austempering temperature (T_A), above the martensitic transformation onset temperature (T_Ms), between 250 and 450 °C, in order to avoid pearlitic transformation.
• Afterwards, material is isothermally kept at T_A, for a certain austempering time (t_A), to allow the formation of an ausferritic matrix. Finally, it is removed from the salt bath and air-cooled to room temperature.

Transformation of ausferritic microstructure takes place upon austempering and is related to the nucleation and growth of ferrite plates within an austenitic matrix. Transformation kinetics is affected by the chemical composition of the melt and heat treatment temperatures as well as the shape, size and distribution of graphite [9]. These austempering phase transformations can be described as a two-stage reaction.

In stage I, known as hardening and represented by reaction (1), the austenite (γ) transforms to a α_ac, which has a lower carbon solubility than austenite, so the excess of carbon atoms in ferrite diffuse into the surrounding austenite. At this stage, ferrite nucleation is triggered by subcooling from T_γ to T_A and happens preferentially at the γ/G interface and austenitic grain boundaries. Thereafter, the growth of this phase occurs in needle-like form promoted by the carbon concentration gradient between α_ac/γ and γ/G interfaces [10]. The incremental rate of α_ac needles is regulated via the rate of diffusion of carbon atoms from the α_ac/γ interface into γ [11]. As transformation proceeds, carbon atoms enrich γ converting it to γ_hC, increasing its stability and avoiding the transition of γ to martensite (α’) during air quenching [12].

γ → α_ac + γ_hC

(1)

At stage II, known as embrittlement and represented by reaction (2), produced when the material is subjected to a long isothermal maintenance, γ_hC slows down α_ac growth as it becomes saturated with carbon; thus, it continues to increase through the precipitation of iron carbides. For short t_A of less than 30 min, ausferritic transformation reaction is not completed, and transformation from γ into α’ occurs. For long t_A of more than 2 h, there exists the risk of reaching Stage II, where precipitation of carbides occurs, deteriorating mechanical properties of the casting [13].

γ_hC → α_ac + iron carbides

(2)

Time elapsed from the end of stage I to the beginning of stage II is known as the processing window [14,15]. The processing window for temperatures close to T_Ms is smaller because kinetics of ausferritic transformation reaction is higher given the elevated carbon saturation of austenite, which is a consequence of a greater α_ac needle nucleation [16].

Time and temperature of heat treatment modify phase transformations occurring through austempering since during the ausferritic transformation reaction shearing takes place, in addition to carbon atom diffusion [17]. T_A affects mainly the nucleation of α_ac needles and the diffusion capacity of carbon atoms. For a T_A near T_Ms a higher α_ac needle nucleation occurs. Nevertheless, carbon diffusion is limited due to the low temperature resulting in an ADI with high hardness and mechanical strength but weak ductility. Meanwhile, at higher temperatures, α_ac needle nucleation decreases but carbon diffusion is favored [9,18], leading to an ADI with high ductility and poor mechanical strength [19,20].

Industrial use of ADI is complicated when controlling and establishing the variables of production processes, as part and heat treatment design requires a large number of experimental tests in order to validate successful production. Thus, the study of new processing methods, e.g., two-step austempering and usage of mathematical models for predicting final properties and microstructure will contribute to reducing the uncertainty associated with the production, development and implementation of new applications.

Data interpretation has become an important task to be considered within research and development activities of new products. Tools such as the statistical data-driven model (SDDM) enable the prediction and discovery of the repercussions of certain processing variables involved in manufacturing and experimental processes for the validation of prototypes or new products on certain characteristics or properties of interest [21,22,23,24]. This allows a better approach to the design of experiments (DOE) and the search for process optimization based on prediction by statistical analysis. Some related works relying on the SDDM are presented below.

Sivaiah et al. [25] proposed the selection of optimal cutting conditions for 17-4 PH stainless steel turning using the Taguchi method. From statistical analysis results and considering the impact of cutting speed and feed rate, depth of cut and type of coolant, the authors concluded that the optimization of this machining technique is absolutely sustainable, reducing environmental footprint and material waste and enhancing productivity. Montes et al. [26] applied a statistical mathematical model to establish the effect of rotational speed and feed rate on corrosion resistance for dissimilar Al-Cu welds by FSW. By means of this model they determined there is a greater change in corrosion rate as feed rate increases, established the parameters to acquire good quality joints and outlined a microstructural study of their repercussions. Application of statistical analysis in industrial processes such as welding was also carried out by Chandran et al. [27], who proposed laser beam welding parameters with direct influence on tensile and hardness values based on analysis of variance (ANOVA). Analogous to the aforementioned researchers, Choi [28] performed the same kind of statistical analysis method to evaluate the result of using different thicknesses of AZ31 magnesium alloy parts on fatigue crack growth. The author concluded that this procedure could ascertain the impact of part thickness on the characteristics of crack growth as well as on the propagation velocity. Another industrial process implementing ANOVA is parameter optimization in the powder metallurgy process of Al-MoO₃ composites, where Sudha et al. [29] sought to obtain the lowest corrosion rate in combination with high compressive strength. They discovered MoO₃ reinforcement content has a considerable effect on compressive strength, and corrosion resistance is mostly affected by sintering temperature.

In addition to such statistical studies, artificial intelligence techniques such as fuzzy logic and artificial neural networks are an option for process data analysis as reviewed by Kalpanapriya [30]. These approaches have recently been employed to predict the hardness values in ADI [31,32].

This research provides a novel approach to microstructural studies and their effect on hardness for ADI supported by the statistical data-driven model (SDDM). In this case, ANOVA determines the significant degree of the input variables (time and temperature) for ADI hardness. In addition, if the upper and lower confidence intervals are calculated narrowly, the results of each experiment can be statistically replaced with the mean in each configuration.

Therefore, the prediction of the hardness in ADI is represented by a numerical approximation. The model estimates the hardness for the heat treatments with data experimentation presented in this research, and even it can deal with values outside the DOE domain. In addition, the proposed regression model requires fewer data to predict the hardness in comparison to artificial neural network (ANN) techniques, as the ANN needs an extensive database and a high cost of computation for the training stage. On the other hand, fuzzy logic estimation is based on the human experience; therefore, a unique model for the process is complicated to establish.

2. Materials and Methods

2.1. Heat Treatments

Three samples per heat treatment condition of 20 mm × 20 mm × 10 mm were obtained from a ductile iron profile provided by the company Questum Castings (Santa Catarina, Mexico) of class 100-70-03, according to ASTM A536 [33], with the chemical composition described in

Table 1. Chemical composition (wt.%) of ductile iron.

C	Si	Mn	P	S	Cr	Cu	Sn	Mg	Fe
3.250	2.600	0.850	0.018	0.013	0.080	0.710	0.015	0.053	92.411

.

Two muffle furnaces were utilized for austenitizing and austempering heat treatments of ductile iron, a Thermo Scientific FD1535M (Facultad de Ingeniería, Arteaga, Mexico) for austenitizing, carried out at 950 °C for 120 min, and a TL-EMISOR furnace (Facultad de Ingeniería, Arteaga, Mexico) for austempering, varying the temperature every 30 °C from 290 to 380 °C for 30, 60, 90 and 120 min, and finally cooling to room temperature. A combination of potassium nitrate (KNO₃) and sodium nitrate (NaNO₃) salts in a 1:1 ratio was used for austempering heat treatment. Figure 1 illustrates the schematic diagram of heat treatments performed to obtain ADIs.

2.2. Microstructural Characterization

On completion of heat treatments, samples were prepared metallographically. Microstructural characterization of ductile iron, before and after heat treatments, was conducted under an Olympus optical microscope (I.T. Saltillo, Saltillo, Mexico) and ImagePro image analyzer (I.T. Saltillo, Saltillo, Mexico). Samples were etched in a solution of nitric acid and 2% ethyl alcohol at room temperature, for approximately 10 s, to reveal microstructure. To quantify the γ_hc of each sample, 10 micrographs were taken at 500× magnification to guarantee the reliability of results.

2.3. Hardness Tests

Rockwell C hardness measurements were made on a Buehler Macromet 5100 and a diamond tip indenter, under a load of 150 kgf for 10 s on the previously polished surface, as per ASTM E18 standard specification [34], with a distance of 3 mm between each indentation, extending from the center to both sides of the material. A total of 15 hardness measurements were performed for each heat treatment condition.

3. Results and Discussion

3.1. Microstructural Analysis and Hardness

According to the chemical analysis developed by the company Questum Castings, presented in Table 1, C and Si contents are within the recommended composition for ductile iron; notwithstanding, Mn content is slightly high, ranging from 0.2 to 0.7%. While adding this element improves mechanical properties without affecting the machining process, there is a risk of carbide formation for contents above 0.7%. Notably, Mn has a very severe positive segregation, so the addition of copper helps to counteract it. Further, Mn and Cu together enhance the hardenability of the castings, preventing the occurrence of pearlite, which reduces mechanical properties, during severe quenching in the salt bath [35,36].

The microstructure of the metal matrix of ductile iron prior to the austempering heat treatment is completely pearlitic, as shown in Figure 2. This is attributed to the addition of Cu and Mn in ductile iron promoting the appearance of the pearlitic matrix, and together with Sn, the transformation to pearlite is ensured during solidification because it reduces the minimum subcooling for pearlite nucleation [37,38,39]. The nodularity is 96%, and nodule count equals 90 nodules/mm². The total volume of graphite is 10.33%, and that of the metal matrix is 89.67%.

Microstructures obtained after austempering heat treatments exhibit a characteristic ausferritic metallic matrix composed of α_ac needles and γ_hC phase, as seen in Figure 3.

Morphology, distribution, and size of ausferrite depend on the site where nucleation occurs, carbon saturation of austenite during growth and austempering temperature. Figure 4 depicts the different morphologies of the phases found in some samples. Figure 4a shows for the sample treated at 320 °C for 30 min, in a faint gray color, the appearance of α’ between the α_ac needles. This happens even though the temperature is above the martensitic transformation starting one, since austenite is not completely enriched with carbon, leaving some untransformed austenite, which upon cooling to room temperature is transformed into martensite, in addition to material internal stresses [12]. In Figure 4b it can be noted how in the sample treated at 290 °C for 90 min, α_ac needles converge at the same nucleation site, giving the appearance of α_ac needle bundles. Another type of α_ac needle arrangement consists of clusters, as observed in the sample treated at 320 °C for 90 min in Figure 4c, which is composed of parallel α_ac needles separated by γ_hC, similar to results of other researchers [40,41,42,43]. Meanwhile, Figure 4d,e illustrate the two types of α_ac needles, fine (lower ausferrite) and feathery (upper ausferrite), respectively, within the samples treated at 290 and 380 °C for 60 min, as reported by Putantunda and Olawale [19,44].

The microstructures obtained under all treatment conditions are shown in Figure A1 in Appendix A, demonstrating an increase in the presence and size of γ_hC blocks caused by an increment in T_A. This behavior has been also documented in several investigations and is attributed to carbon atoms diffusing longer distances and stabilizing a greater amount of austenite, resulting in lower nucleation of α_ac needles [45,46,47]. The T_A effect is also evidenced by this figure as it is possible to observe the enlargement and thickening of α_ac needles as time elapses, with a redistribution of carbon and achieving a transformation of γ_hC into α_ac [19,48,49]. Moreover, for the same t_A, ferrite inserts become thick when produced at higher temperatures given that the austempering process is highly dependent on the rate of carbon diffusion.

Change in ausferritic morphology is exhibited in Figure 5. At a temperature of 290 °C (Figure 5a) the presence of lower ausferrite indicated by red circles is evident, at 320 °C (Figure 5b) a combination of lower and upper ausferrite can be observed and finally at temperatures of 350 and 380 °C (Figure 5c,d) an all upper ausferritic morphology is indicated by yellow circles. Such a transition in ausferritic morphology is primarily influenced by T_A and occurs because at temperatures around T_Ms the transformation is dominated by α_ac needle nucleation, and at higher temperatures the transformation is dominated by their growth, as reported by [19,45,50].

Throughout the austempering process, cooling is performed from T_γ to T_A with its impact being observed in terms of α_ac needle nucleation. Figure 6 depicts the microstructural outcome of this cooling, which demonstrates that as T_A goes up, there is a limit in α_ac nucleation and, therefore, a rise in γ_hC stabilization. At temperatures ranging from 290 to 350 °C, regions that have not been enriched with sufficient carbon to prevent the transformation from γ to α’are detected. As T_A increases, the formation of these zones decreases until reaching 380 °C, where they disappear since the diffusion of carbon atoms is facilitated by the effect of temperature. For all 30 min T_As, α’ is found between the α_ac needles.

Figure 7a shows the γ_hC quantification for the different heat treatment conditions. It is observed that increasing T_A increases the amount of γ_hC, for a constant time. On the other hand, as t_A increases at the same T_A there is a maximum in γ_hC content after which diffusional transformation of γ_hC to α_ac is favored. These results are similar to those found by other researchers both in ADI [51,52] and in steel subjected to austempering treatment [53].

The lowest level of γ_hC is at 30 min of treatment, attributed to the γ low carbon stabilization, so during post-austempering air-cooling, a large part of this γ transforms into α’ [12], as exposed in Figure 6. After 60 min a decrease in γ_hC is noticed for temperatures of 290 and 320 °C, and for 350 °C this reduction takes place until 120 min, while at 380 °C γ_hC content stabilizes. This behavior derives from transformation kinetics, i.e., as the T_A diminishes, the ausferritic transformation reaction proceeds faster, consuming γ_hC for α_ac formation and at possible carbide precipitation [54,55,56]. Figure 7b shows hardness values decreasing as T_A increases, consequence of a higher amount of γ_hC and the change of ausferritic morphology in the metal matrix. A decrease in hardness can also be noticed as t_A increases, given that carbon redistribution and coarsening of ausferritic microstructure are present [48,49].

3.2. Mathematical Model

This section introduces the SDDM evaluation of hardness values retrieved from experimentation. In the first analysis, an ANOVA was performed to determine t_A and T_A effect on sample hardness. For the second analysis, a regression model was used to numerically describe the process and predict hardness values for each t_A and T_A condition. The third analysis determined the quality of both the proposed regression model and the experimental data through the residual error between them. Experimental data and results must satisfy the following assumptions:

Assumption 1.

The probability distribution (Students’s t) of analyzed data is a normally distributed distribution for small experimental sample sizes as is the case of ADI studied in this research.

Assumption 2.

The alternative hypothesis from ANOVA validates the significance impact of t_A and T_A on hardness degree of the samples.

Theorem 1.

If the number of samples for each experiment is n < 30 and the confidence intervals in a Student’s t distribution with a significance level α = 0.05 are narrow, then the number of experiments to be statistically analyzed is reduced to the mean $\overline{y}$ of each experimental setup.

Proof of Theorem 1.

Two-tailed confidence intervals with a probability $\frac{\mathsf{\alpha }}{2}$, using the $\mathrm{t}$-distribution for samples smaller than 30 are calculated according to Equation (A1). For the 16 experimental setups in

Table A1. Results of the statistical analysis for the proposed hardness estimation regression model.

TA (°C)	${\mathit{t}}_{\mathit{A}}\left(\mathbf{min}\right)$	Hardness (Mean)	Sample Std. Dev.	Lower Limit (Li)	Upper Limit (Ui)
290	30	52.88	0.7043	52.4900	53.2700
	60	47.69	0.5768	47.3673	48.0061
	90	46.73	0.4131	46.4979	46.9554
	120	47.25	0.5276	46.9545	47.5389
320	30	48.26	0.8806	47.7723	48.7477
	60	44.36	0.7944	43.9201	44.7999
	90	44.09	0.6933	43.7094	44.4773
	120	43.44	0.7366	43.0321	43.8479
350	30	44.84	0.7744	44.4111	45.2689
	60	39.69	0.7588	39.2651	40.1056
	90	37.53	1.0055	36.9765	38.0901
	120	36.78	0.9930	36.2301	37.3299
380	30	41.78	0.6879	41.3991	42.1609
	60	37.89	0.6717	37.5147	38.2587
	90	34.65	0.6093	34.3159	34.9907
	120	33.67	0.6230	33.3217	34.0117

, 15 hardness tests each were performed, meaning a total of 240 experiments. Given n = 15 and the distribution random variable $\mathrm{t}=2.1448$, the upper and lower bounds are found with a 95% confidence level, i.e., t ($\mathsf{\alpha },\mathrm{n})$. Table A1 sets out the confidence limits quite close to the mean ${ \mathrm{U}}_{\mathrm{i}}{\mathrm{L}}_{\mathrm{j}}~\overline{\mathrm{y}}$. All this denotes an acceptable experimental design enabling the report of 240 experiments to be statistically replaced with the mean at each of the 16 configurations. Details of can be seen in Appendix B. □

Figure 8 depicts the normal probability of average hardness versus the percentile sample of each test. A 97.13% linear trend across the distribution suggests an acceptable experimentation for statistical analysis.

3.3. ANOVA

Based on Figure 8 it is possible to implement a Pearson correlation analysis $\left(r\right)$ to examine the relationship between variable x (time and temperature) and variable y (hardness), involving $-1\le r\le 1$. If $r>0$, $“y”$ tends to increment as $“x”$ increments; if $r<0$, and $“y”$ tends to decrement as $“x”$ increments. The stronger the linear relationship of $“x”$ and $“y”$, the closer r is to −1 or 1; the weaker the linear relationship, the closer it is to 0. This means that the inversely proportional behavior from t_A and T_A causes a higher effect of the temperature on the hardness values in comparison to the time effect. This can also be clearly observed in the microstructural results, where the T_A modifies the ausferritic morphology, and the t_A affects the ausferritic transformation progress. Therefore, the hardness values need to be adjusted according to quality standards requirements. As a consequence, tempering temperatures are recommended. This coefficient is estimated as in Equation (3) and results from

Table 2. Pearson correlation matrix between time and temperature variables for hardness.

Variables	Temperature (°C)	Time (min)	Hardness
Temperature (°C)	1.0000	0.0000	−0.8511
Time (min)	0.0000	1.0000	−0.4573
Hardness	−0.8511	−0.4573	1.0000

and

Table 3. Type of relationship and increases.

Variables	Relationship	Increase
Hardness–Temperature	Strong	Negative
Hardness–Time	Weak	Negative
Temperature–Time	Null	Null

, which list the temperature–time and ADI hardness Pearson correlation, that is, the type of relationship between variables and the kind of increase among them, respectively, in agreement with [56,57].

$$r=\frac{s_{xy}}{s_x{s}_y}$$

(3)

where $s_x$ and $s_y$ are the standard deviations, and $s_{xy}$ the covariance of $“x”$ and $“y”$, correspondingly [57].

ANOVA was conducted to identify the influence of one or more factors on the mean hardness, i.e., whether or not they have a significant effect on the experiment. Similarly, sampling theory seeks to compare the dissimilarities of two or more sample means, based on the assumption that populations have the same variance. It is often necessary to test the hypothesis which assumes that means are equal [56,58]. The significance level α is the probability of rejecting the null hypothesis when it is true and helps us to establish whether the contrast statistic is in the rejection zone or not, for this study and as mentioned in Theorem 1, α = 0.05.

The null hypothesis ($H_0$) points to the absence of significant differences, produced by the variables, in the phenomenon. This is accepted if:

$$F_{calculated}<F_{critical}$$

The alternative hypothesis ($H_i$) states that significant differences do exist in the phenomenon produced by the variables. This hypothesis is accepted if:

$$F_{calculated}>F_{critical}$$

$F_{critical}$ is attained from the Fisher F distribution conditioned on the degrees of freedom. The statistical factor between sample variances $S_1^2/S_2^2$ is considered in this distribution. When ratio is large or small it indicates a huge difference across the variances; conversely, when near 1 it indicates a small difference between analyzed samples. If we have two samples of size $N_1$ and $N_2$, $S_1$ and $S_2$ represent the sample variances, respectively, taken from two normal populations with variances ${\sigma }_1{}^2$ and ${\sigma }_2{}^2$. $F$-statistic is defined as in Equation (4):

$$F_{calculated}=\frac{\raisebox{1ex}{${\widehat{S}}_1{}^2$}\!\left/ \!\raisebox{-1ex}{${\sigma }_1{}^2$}\right.}{\raisebox{1ex}{${\widehat{S}}_2{}^2$}\!\left/ \!\raisebox{-1ex}{${\sigma }_2{}^2$}\right.}, {\widehat{S}}_n{}^2=\frac{N_n{S}_n{}^2}{N_n-1}$$

(4)

Fisher’s F distribution degrees of freedom are calculated as $v_n=N_n-1$.

Such a Fisher statistical distribution is given by Equation (5) according to [50,52]

$$T=\frac{C{F}^{\left(\frac{v_1}{2}\right)-1}}{{\left(v_{1}F+v_2\right)}^{\left(v_1+v_2\right)/2}}$$

(5)

with $C$ being a constant dependent on $v_1$ and $v_2$, so the area under the curve equals 1. In the distribution tables, $v_1$ stands for the degrees of freedom in the numerator ($v_R$ y $v_C$) and $v_2$ for those in the denominator ($v_E$) [56,58]. Assuming population variances ${\sigma }_1$ $={\sigma }_2$ or $H_0$, $F_{calculated}$ is obtained from:

$$\raisebox{1ex}{$S_R{}^2$}\!\left/ \!\raisebox{-1ex}{$S_E{}^2$}\right. \mathrm{and} \raisebox{1ex}{$S_C{}^2$}\!\left/ \!\raisebox{-1ex}{$S_E{}^2$}\right.$$

(6)

Variation among treatments is as follows:

$$V_R=b{\displaystyle \sum }_{i=1}^a{\left({\overline{X}}_i-{\displaystyle \stackrel{̿}{X_T}}\right)}^2$$

(7)

where $a$ is the number of rows, $b$ the number of columns,${\overline{X}}_i$ the average of the rows and $\stackrel{̿}{X_T}$ the total average. Block-to-block variation is described as:

$$V_C=a{\displaystyle \sum }_{i=1}^b{\left({\overline{X}}_j-{\displaystyle \stackrel{̿}{X_T}}\right)}^2$$

(8)

where ${\overline{X}}_j$ is the average of the columns. The total data variance is defined as:

$$V={\displaystyle \sum }{\left(x_{i,j}-{\displaystyle \stackrel{̿}{X_T}}\right)}^2$$

(9)

with $x_{i,j}$ representing the values from Table A1. Therefore, the residual variance of collected data becomes:

$$V_E=V-V_R-V_C$$

(10)

The mean square between treatments is:

$$S_R{}^2=\frac{V_R}{a-1}$$

(11)

while inter-block mean square equals:

$$S_C{}^2=\frac{V_C}{b-1}$$

(12)

and the mean square between residuals is expressed:

$$S_E{}^2=\frac{V_E}{\left(a-1\right)\left(b-1\right)}$$

(13)

in which $S_R=S_{Temperature}$ and $S_C=S_{Time}$.

From

Table 4. Results of the analysis of variance across factors of time and temperature affecting hardness.

Variable	Variation (Sum of Squares) Quadratic Sum	$\mathit{d}\mathit{o}\mathit{f}$	Mean Quadratic	${\mathit{F}}_{\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$	Probability	${\mathit{F}}_{\mathit{c}\mathit{r}\mathit{i}\mathit{t}\mathit{i}\mathit{c}\mathit{a}\mathit{l}}$
Temperature	328.47185	3.00	109.49062	122.55530	0.0000001	3.86255
Time	110.62773	3.00	36.87591	41.27603	0.0000138	3.86255
Error	8.04058	9.00	0.89340
Total	447.14016	15.00

, $H_1$ is validated for ADI treatment results, meaning that $F_{calculated}$ $\ge$ $F_{critical}$ for T_A and t_A, i.e., there is an effect of these two variables on sample hardness degree.

Probability displayed in Table 4 variance analysis comes from the right tail of $F$ distribution corresponding to the $F_{calculated}$ value for treatments and blocks with their respective degrees of freedom. This probability is computed via the Excel function distr.F.CD ($F_{calculated}$, treatment or block degrees of freedom ($dof$), error degrees of freedom).

3.4. Least Squares Regression

The model to approximate the hardness value in terms of the austempering time and temperature is obtained through a power equation based on the least square method as is presented below:

Corollary 1.

When the analyzed data set exhibits substantial errors, the use of polynomial interpolation is inappropriate in view of the unsatisfactory prediction level at intermediate values. In contrast, an approximation of a general trend of the data, by a power regression, is more useful for minimizing the sum of residual squared errors between $y{ }_{measured}$ and $y{ }_{calculated}$, estimated with the model.

The squared error sum is presented in Equation (14) as reported in [58]:

$$S_r={\displaystyle \sum }_{i=1}^n{e}_i^2={(y_{, measured}-y_{calculated})}^2$$

(14)

A regression model with a power equation is proposed as in Equation (15), in which a hardness function in terms of temperature and time is provided for ADI heat treatment:

$$D=a_0{T}_A{}^{a_1}{t}_A{}^{a_2}\to \ln \left(D\right)=\ln \left(a_0\right)+a_1\ln \left(T_A\right)+a_2\ln \left(t_A\right)$$

(15)

where $a_0$,$a_1$, $a_2$ are the coefficients for describing the function of this process and $D$ is the hardness. Using the properties of natural logarithms, Equation (15) can be linearized. In a multiple regression, the linearized power equations are the most suitable to fit experimental data. Upon applying Equations (14) and (15), the sum of the quadratic errors is determined for the particular case of the study carried out in this research.

$$S_r={\displaystyle \sum }_{i=1}^n{e}_i{}^2={\displaystyle \sum }_{i=1}^n{\left[\ln \left(D_i\right)-\ln \left(a_0\right)-a_1\ln \left(T_A{}_i\right)-a_2\ln \left(t_A{}_i\right)\right]}^2$$

(16)

To compute the coefficient values, Equation (16) is derived with respect to each coefficient, recalling that $\frac{\partial S_r}{\partial a_0},\frac{\partial S_r}{\partial a_1}, \frac{\partial S_r}{\partial a_2}$ must equal zero to minimize the error between measured and calculated data in Equation (14). Once the parameters are estimated, the following model emerges:

$$D=e^{10.393}{T}_A{}^{-1.0599}{t}_A{}^{-0.1173}$$

(17)

3.5. Quantification of Error in Regressions

In accordance with [57,59], the magnitude of residual error associated with the dependent variable before regression is

$$S_t={\displaystyle \sum }_{i=1}^n{(y_{, measured}-y_{mean})}^2$$

(18)

where S_t is the total sum of squares around dependent variable $‘y’$ mean. Subsequent to regression, $S_r$ is estimated, which represents the sum of squares of the residuals around the regression function. This characterizes the residual error remaining after regression. The difference between these two summations, $S_t-S_r$, quantifies error through data description in terms of a straight line rather than an average value. Because the magnitude of this quantity depends on the scale, the difference is normalized to $S_t$ to obtain:

$$r^2=\frac{S_t-S_r}{S_t}$$

(19)

Here, r² is known as the coefficient of determination, and r is the correlation coefficient ($\sqrt{r^2}$). A perfect data fit, S_r = 0 and r = r² = 1, indicates that function explains 100% of data variability. If r = r² = 0 and S_t = S_r the fit does not imply any improvement.

Standard error is formulated as in Equation (18). Said number is divided by $n-\left(m+1\right)$ since ($m+1$) coefficients drawn from the data, $a_0,a_1,\dots \dots \mathrm{\dots },a_m$, were employed to calculate $S_r$; $m+1$ degrees of freedom are lost. Briefly it can be stated that:

$$S_{y/x}=\sqrt{\frac{S_r}{n-\left(m+1\right)}}$$

(20)

where m + 1 = total degrees of freedom and n = number of experiments.

Adjusted coefficient of determination $r^2{}_{adjusted}$ is used in multiple regression to observe the intensity or effectiveness of independent variables (T_A and t_A) with regard to the dependent one ($D$). The computation of $r^2{}_{adjusted}$ is set out in Equation (21):

$$r^2{}_{adjusted}=1-\frac{n-1}{n-m+1}\left(1-r^2\right)$$

(21)

Table 5. Results of the statistical analysis for the proposed hardness estimation regression model.

Regression Statistics	Values
$\mathit{r}$	0.9757
${\mathit{r}}^{\mathbf{2}}$	0.9520
${\mathit{r}}^{\mathbf{2}}{}_{\mathit{a}\mathit{d}\mathit{j}\mathit{u}\mathit{s}\mathit{t}\mathit{e}\mathit{d}}$	0.9446
${\mathit{S}}_{\mathit{y}\mathbf{/}\mathit{x}}$	1.2845
n	16

reveals the direct relationship of hardness degree and the magnitude of ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{t}}_{\mathrm{A}}$. Based on the coefficient of determination r², it is established that the proposed power equation model is able to describe 95.2% of the phenomenon uncertainty. As per $r^2{}_{odjusted}$, contemplated variables occupy 94.46% of model effectiveness, which means that there are some other factors that impact the remaining 5.54% of the phenomenon. The power equation model has an estimation error of ±1.2845.

Theorem 2.

If the variability of the experimental data r² is ~$1$and standard error converges towards zero $S_{y/x}\to 0$, then the upper ($U$) and lower ($L$) interval values of the power regression coefficients$a_0$,$a_1$,$a_{2 }$in Equation (15), evaluated at a 95% confidence level, converge to values near each other in very close intervals.

Proof of Theorem 2.

For the regression model (17), the coefficient of determination ${ \mathrm{r}}^2=95.20\%$, the standard error ${ \mathrm{S}}_{\mathrm{y}/\mathrm{x}}=1.2845$ and the values of upper $\mathrm{U}$ and lower $\mathrm{L}$ intervals are determined by Equations (A5) and (A6), Results are listed in

Table A3. Linear and power model comparison in a random domain.

i	T (°C)	t (min)	Powers	Linear	Difference
1	350	168	35.9894	33.8796	2.1098
2	429	52	33.2836	31.6430	1.6406
3	301	74	46.4906	47.2279	0.7373
4	382	58	37.1604	37.5154	0.3550
5	424	58	33.2709	31.8813	1.3896
6	442	81	30.6134	27.8089	2.8045
7	213	36	72.9883	61.7717	11.2166
8	246	125	54.1428	50.9300	3.2127
9	358	47	40.8000	41.5277	0.4839
10	331	145	38.8478	38.0861	0.6808

for each coefficient ${\mathrm{a}}_0$,${\mathrm{a}}_1$, ${\mathrm{a}}_2$. Details can be noticed in Appendix C. □

Remark 1.

On the basis of Theorem 2, the Equation (17) model holds for increasing numbers of experimental data and in domains outside tested data. This prevents excessive experimentation.

Figure 9 compares the experimental results and those estimated by the model demonstrating the impact of ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{t}}_{\mathrm{A}}$ on hardness. Model results are reliable at least in the experimental range.

Figure 10 depicts the surface simulation of the data reconstructed by the proposed model for each ${\mathrm{T}}_{\mathrm{A}}$ and ${\mathrm{t}}_{\mathrm{A}}$ hardness value. For the domain 30 < ${ \mathrm{t}}_{\mathrm{A}}$ < 60 and 290 < ${ \mathrm{T}}_{\mathrm{A}}$ < 315, the highest hardness values are attained, ascribed to the presence of α’ and lower ausferrite, respectively, as 53.78 HRC was yielded from the application of a treatment at 290 °C for 30 min. In the case of the domain 70 < ${ \mathrm{t}}_{\mathrm{A}}$ < 120 and 340 < ${ \mathrm{T}}_{\mathrm{A}}$ < 380, the lowest hardness values are achieved, a consequence of a coarsening of the microstructure and a higher amount of γ_hC, correspondingly. In other words, the higher the temperature, the lower the hardness and vice versa, and the longer the treatment time, the lower the hardness and vice versa. This behavior has also been documented by other authors [60,61,62], and these materials exhibited excellent wear resistance and tensile strength properties for their possible application as high-performance components.

4. Conclusions

• The effect of austempering time and temperature on hardness shows an inversely proportional behavior, which was verified by the SDDM. This relationship is attributed to the microstructural characteristics acquired during the heat treatment.
• ANOVA quantifies the statistical influence of the time and temperature of tempering on the degree of hardness for ADI samples. The correlation study shows there is a strong inverse relationship between hardness and temperature, i.e., this is consistent with the microstructural characteristics found in the treatment. In the power regression model indicates that the multiple correlation between variables is around 97.57%, the adjusted coefficient of determination is 94.46% and the standard error is ±1.2845.
• By means of ANOVA, it was possible to quantify the significant influence of austempering time and temperature on the hardness degree of ductile iron samples. Multiple correlation among variables is about 97%. The correlation study shows there is a strong inverse relationship between hardness and temperature, i.e., the higher temperature lowered the hardness and vice versa. This is consistent with the microstructural characteristics found in the treatment.
• A model representing the numerical response of hardness degree as a function of time and temperature in the austempering treatment of ductile irons is presented. Model variability is 95.20% in the face of uncertainties, and according to Theorem 2, the proposed model can avoid excessive experimentation for this process. Additionally, quality of the data and model was analyzed, and confidence intervals were defined for both cases.

Future work including the effect of alloying elements on hardness in a new mathematical model is being considered.