1. Introduction

2. Materials and Methods

3. Experimental Results and Discussion

3.3. Effect of heat treatment on mechanical characteristics: planned experiment and optimization

According to the one-factor-at-a-time experimental results, the governing factors were chosen to change as follows: ${200}^{°}\mathrm{C}\le \mathrm{T}\le {700}^{°}\mathrm{C}$ and $1\mathrm{h}\le \mathrm{t}\le 4\mathrm{h}$. Table 2 lists the governing factor levels. The correlation between natural ${\tilde{\mathrm{x}}}_{\mathrm{i}}$ and coded ${\mathrm{x}}_{\mathrm{i}}$ coordinates is

$${\mathrm{x}}_{\mathrm{i}}=\frac{2\left({\tilde{\mathrm{x}}}_{\mathrm{i}}-{\tilde{\mathrm{x}}}_{0,\mathrm{i}}\right)}{{\tilde{\mathrm{x}}}_{\max ,\mathrm{i}}-{\tilde{\mathrm{x}}}_{\min ,\mathrm{i}}},$$

(1)

where ${\tilde{\mathrm{x}}}_{\max ,\mathrm{i}}$, ${\tilde{\mathrm{x}}}_{0,\mathrm{i}}$ and ${\tilde{\mathrm{x}}}_{\min ,\mathrm{i}}$ are the upper, middle, and lower levels of the ith factor in natural coordinates, respectively.

The objective functions are the following mechanical characteristics: yield limit (${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}$), tensile strength (${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}$), elongation (${\mathrm{Y}}_{{\mathrm{A}}_5}$), hardness (${\mathrm{Y}}_{\mathrm{H}\mathrm{B}}$) and impact toughness (${\mathrm{Y}}_{\mathrm{I}\mathrm{T}}$). Figure 18 provides the experimental points in the plane of governing factors. Table 3 lists the experimental outcomes for the chosen mechanical characteristics.

An analysis of variance (ANOVA) was conducted via QStatLab [10] to investigate the significance of the governing factors. Figure 19 provides the main ANOVA effects. For all objective functions, the more significant factor is ${\mathrm{x}}_1$ (temperature). Time has the most substantial influence on the tensile strength. The yield limit (Figure 19a) is maximum when the temperature is at the middle level and the time is at the second level (${\mathrm{x}}_2=-0.5$). The combination of maximum temperature and time at the middle level minimises the yield limit. The influence of the governing factors on the tensile strength is similar (Figure 19b). When the temperature and time simultaneously occupy the fourth level $\left({\mathrm{x}}_1={\mathrm{x}}_2=0.5\right)$, the elongation is maximal (Figure 19c). The combination of the minimum time and temperature at the second level (${\mathrm{x}}_1=-0.5$) minimises the elongation. When the temperature is at the second level $\left({\mathrm{x}}_1=-0.5\right)$ and the time is at the fourth level $\left({\mathrm{x}}_2=0.5\right)$, the hardness is maximum (Figure 19d). The minimum hardness is obtained when the temperature is maximum and the time is at the middle level $\left({\mathrm{x}}_2=0\right)$. The influence of the governing factors on the impact toughness is analogous to their influence on the elongation (Figure 19e). The ANOVA predicts the influence of the governing factors only in a qualitative aspect. More accurate results are obtained after mathematically modelling the studied mechanical characteristics.

The experimental results for the mechanical characteristics were subjected to regression analyses. The significance of the regression coefficients was determined at the $\mathrm{p}=0.05$. Given the chosen experimental design (five levels for each factor), the approximating polynomials may be of degree 4 or lower:

$${\mathrm{Y}}_{\mathrm{k}}\left(\left\{\mathrm{X}\right\}\right)={\mathrm{b}}_0+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}-1}{\displaystyle \sum _{\mathrm{j}=\mathrm{i}+1}^{\mathrm{m}}{\mathrm{b}}_{\mathrm{i}\mathrm{j}}{\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}}}}}+{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{b}}_{\mathrm{i}\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}^2+\dots },\mathrm{k}=1,2,\dots \mathrm{q},\dots ,$$

(2)

where $\left\{\mathrm{X}\right\}$ denotes the vector of the governing factors, $\mathrm{m}$ represents the number of governing factors, and $\mathrm{q}$ indicates the number of objective functions.

The regression analyses were performed using QStatLab [10], and Table 4 presents the regression coefficients. The magnitude (absolute value) of the coefficients in front of the dimensionless variables indicates the significance of the corresponding governing factor (variable), and the absolute value of the coefficients in front of the products of the variables indicates the significance of the interaction between the governing factors. The regression coefficients in Table 4 indicates that: 1) the ageing temperature is a much more significant factor than the ageing time, confirming the ANOVA results, and 2) the interaction between the governing factors is relatively weak, with the exception of the tensile strength.

Table 3 presents the values of the objective functions calculated using Eq. (2) for the experimental points from the plan. The comparison between the experimental results for the objective functions and those predicted by the models (at the experimental points) displays excellent agreement. Figure 20 presents a graphical visualisation of the models. The type of surfaces confirms that the ageing temperature is the more significant of the two factors. The ageing time influences the tensile strength most strongly (confirming the ANOVA results), whereas the influence is weakly expressed for the other characteristics. The factor least sensitive to the ageing time is the impact toughness.

The two primary characteristics of static strength (yield limit and tensile strength) similarly depend on the temperature. As the temperature increases, the static strength increases and reaches its maximum value between 400°C and 500°C, after which it begins to decrease at a faster rate. The behaviour of hardness is similar, but it reaches its maximum values earlier (in the interval between 250°C and 300°C) and then decreases to a minimum. The elongation and the dynamic strength (impact toughness) display similar behaviour under temperature and time changes because both characteristics have a common physical basis. The behaviour of the dynamic strength when changing the temperature is opposite that of the static strength. The maximum values of all objective functions, $\max {\mathrm{Y}}_{\mathrm{j}}$, and their corresponding magnitudes of the governing factors, ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{m}}$, were found with QStatLab using the random search method with 1,000 iterations. Table 5 lists the results.

Table 5. Maximum values of the objective functions and the corresponding governing factors.

Table 5. Maximum values of the objective functions and the corresponding governing factors.

Objective fun ctions	Governing factors		$\mathbf{Maximum}\mathbf{values}\mathbf{max}{\mathit{Y}}_{\mathit{j}}$
	Codded	Natural
${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}},\mathrm{M}\mathrm{P}\mathrm{a}$	${\mathrm{x}}_1^{\mathrm{m}}=$-0.19582 ${\mathrm{x}}_2^{\mathrm{m}}=$0.98008	${\mathrm{T}}^{\mathrm{m}}={401}^{\mathrm{o}}\mathrm{C}$ ${\mathrm{t}}^{\mathrm{m}}=3\mathrm{h}58\min$	431.8
${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}},\mathrm{M}\mathrm{P}\mathrm{a}$	${\mathrm{x}}_1^{\mathrm{m}}=$0.33077 ${\mathrm{x}}_2^{\mathrm{m}}=$-0.99671	${\mathrm{T}}^{\mathrm{m}}={505}^{\mathrm{o}}\mathrm{C}$ ${\mathrm{t}}^{\mathrm{m}}=1\mathrm{h}$	812.9
${\mathrm{Y}}_{{\mathrm{A}}_5},\%$	${\mathrm{x}}_1^{\mathrm{m}}=$0.75517 ${\mathrm{x}}_2^{\mathrm{m}}=$0.41485	${\mathrm{T}}^{\mathrm{m}}={639}^{\mathrm{o}}\mathrm{C}$ ${\mathrm{t}}^{\mathrm{m}}=3\mathrm{h}7\min$	14.6
${\mathrm{Y}}_{\mathrm{H}\mathrm{B}}$	${\mathrm{x}}_1^{\mathrm{m}}=$-0.68099 ${\mathrm{x}}_2^{\mathrm{m}}=$0.39878	${\mathrm{T}}^{\mathrm{m}}={280}^{\mathrm{o}}\mathrm{C}$ ${\mathrm{t}}^{\mathrm{m}}=3\mathrm{h}6\min$	261.8
${\mathrm{Y}}_{\mathrm{I}\mathrm{T}},\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$	${\mathrm{x}}_1^{\mathrm{m}}=$0.78857 ${\mathrm{x}}_2^{\mathrm{m}}=$0.00032	${\mathrm{T}}^{\mathrm{m}}={647}^{\mathrm{o}}\mathrm{C}$ ${\mathrm{t}}^{\mathrm{m}}=3\mathrm{h}41\min$	73.65

The correlations between the five objective functions were found by eliminating the governing factors for the pair of considered objective functions. These correlations are essential for setting and solving optimisation problems and for correctly defining the functional constraints. The correlations of the hardness with each of the other four mechanical characteristics were determined. Figure 21 graphically visualises the data. The dependencies of the mechanical characteristics on the hardness are nonlinear. As the hardness increases, the static strength increases up to a specific hardness value (approximately 230 HB for the yield limit and 210 HB for the tensile strength), and subsequently decreases. The elongation and dynamic strength trendlines indicates a continuous decrease when the hardness increases.

Four optimization tasks, which have the most significant importance for practice, were formulated and solved:

1)

Maximum plasticity: ${\mathrm{Y}}_{{\mathrm{A}}_5}=\max {\mathrm{Y}}_{{\mathrm{A}}_5}$;

2)

Maximum impact toughness (dynamic strength): ${\mathrm{Y}}_{\mathrm{I}\mathrm{T}}=\max {\mathrm{Y}}_{\mathrm{I}\mathrm{T}}$;

3)

Simultaneous high hardness and static strength: The objective function vector is

$\left\{\overrightarrow{\mathrm{Y}}\left(\left\{\mathrm{X}\right\}\right)\right\}={\left[{\mathrm{Y}}_{\mathrm{H}\mathrm{B}}{\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}{\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}\right]}^{\mathrm{T}}$,

where $\left\{\mathrm{X}\right\}={\left[{\mathrm{x}}_1{\mathrm{x}}_2\right]}^{\mathrm{T}}\in {\mathsf{\Gamma }}_{\mathrm{x}}$, and ${\mathsf{\Gamma }}_{\mathrm{x}}$ is the plane of the governing factors ${\mathrm{x}}_{\mathrm{i}}$. The objective functions must tend to their maximum values: ${\mathrm{Y}}_{\mathrm{H}\mathrm{B}}\to \max {\mathrm{Y}}_{\mathrm{H}\mathrm{B}}$, ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}\to {\mathrm{maxY}}_{{\mathsf{\sigma }}_{\mathrm{y}}}$, and ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}\to \max {\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}$. Based on Figure 21, the following are the functional limitations: ${\mathrm{Y}}_{\mathrm{H}\mathrm{B}}>230\mathrm{H}\mathrm{B}$, ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}>410\mathrm{M}\mathrm{P}\mathrm{a}$, and ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}>750\mathrm{M}\mathrm{P}\mathrm{a}$.

4)

Simultaneously high hardness, static and dynamic strength: The objective function vector is

$\left\{\overrightarrow{\mathrm{Y}}\left(\left\{\mathrm{X}\right\}\right)\right\}={\left[{\mathrm{Y}}_{\mathrm{H}\mathrm{B}}{\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}{\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}{\mathrm{Y}}_{\mathrm{I}\mathrm{T}}\right]}^{\mathrm{T}}$.

The objective functions must tend to their maximum values: ${\mathrm{Y}}_{\mathrm{H}\mathrm{B}}\to \max {\mathrm{Y}}_{\mathrm{H}\mathrm{B}}$, ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}\to {\mathrm{maxY}}_{{\mathsf{\sigma }}_{\mathrm{y}}}$, ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}\to \max {\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}$, and ${\mathrm{Y}}_{\mathrm{I}\mathrm{T}}\to \max {\mathrm{Y}}_{\mathrm{I}\mathrm{T}}$. The following are functional limitations, according to Figure 21: ${\mathrm{Y}}_{\mathrm{H}\mathrm{B}}>190\mathrm{H}\mathrm{B}$, ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{y}}}>320\mathrm{M}\mathrm{P}\mathrm{a}$, ${\mathrm{Y}}_{{\mathsf{\sigma }}_{\mathrm{u}}}>750\mathrm{M}\mathrm{P}\mathrm{a}$, and ${\mathrm{Y}}_{\mathrm{I}\mathrm{T}}>49\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$.

The first two single-objective optimisation tasks require determining the largest value of the corresponding function without functional limitations and satisfying the governing factor limitations (Table 2). Table 5 lists their solution. The last two are multiobjective optimisation problems. The vector $\left\{{\mathrm{X}}_{\mathrm{j}}^{*}\right\}={\left[{\mathrm{x}}_{1,\mathrm{j}}^{*}{\mathrm{x}}_{2,\mathrm{j}}^{*}{\mathrm{x}}_{3,\mathrm{j}}^{*}\right]}^{\mathrm{T}}\in {\mathsf{\Gamma }}_{\mathrm{x}}$ must be determined so that the objective function magnitudes ${\mathrm{Y}}_{\mathrm{k},\mathrm{j}}\left(\left\{{\mathrm{X}}_{\mathrm{j}}^{*}\right\}\right)$ to satisfy the conditions of the corresponding multi-objective optimisation task, and ${\mathrm{x}}_{1,\mathrm{j}}^{*}$, ${\mathrm{x}}_{2,\mathrm{j}}^{*}$, and ${\mathrm{x}}_{3,\mathrm{j}}^{*}$ are the compromised optimal values of the governing factors. The defined multiobjective optimisation tasks were solved by searching for the Pareto-optimal solution approach. The decision was made through the nondominated sorting genetic algorithm (NSGA-II) [11] using QstatLab. A Pareto front offering 50 compromised optimal solutions was obtained for each of the two tasks. Figure 22 and Figure 23 illustrate the Pareto front for the third and fourth optimisation problems. A compromised optimal solution is selected from each Pareto front. Table 6 contains detailed information regarding the solution results for the four optimisation tasks.

The results of the optimizations were experimentally verified. For this purpose, additional samples were manufactured for tensile and impact toughness tests, which were hardened at 920°C in water and subjected to subsequent ageing with the optimal values (Table 6) of the governing factors for the respective optimisation task. The hardness was measured on the impact toughness samples. Each experimental result was obtained as the arithmetic mean of three samples. Table 7 presents the results. The comparison with the theoretical optimisation results displays good agreement.

4. Conclusions

• The primary mechanical characteristics (yield limit, tensile strength, elongation, hardness and impact toughness) of IAB with β-transformation vary widely depending on the governing parameters of the ageing heat treatment. Therefore, they characteristics can be appropriately controlled according to the functional purpose of the corresponding bronze component. Of the two governing factors (temperature and time), the ageing temperature has a significantly greater weight. The temperature interval 640°C to 650°C maximises plasticity and dynamic strength, whereas hardness and static strength reach the maximum value in the interval of 280°C to 500°C.

Figure 23. Generated Pareto front for fourth optimization task: a. impact toughness – hardness; b. tensile strength – hardness; c. yield stress – hardness; d. impact toughness – tensile strength; e. impact toughness – yield stress; f. tensile strength – yield stress.

Figure 23. Generated Pareto front for fourth optimization task: a. impact toughness – hardness; b. tensile strength – hardness; c. yield stress – hardness; d. impact toughness – tensile strength; e. impact toughness – yield stress; f. tensile strength – yield stress.

• Four optimisation tasks, with the most significance in practice, were formulated and solved. Thus, the optimal (compromise optimal) values of the temperature and time and the corresponding optimal (compromise optimal) magnitudes of the mechanical characteristics for the respective optimisation task were obtained.
• The correlations of the hardness with each of the other four mechanical characteristics were determined. The dependencies of the mechanical characteristics on the hardness are nonlinear. As the hardness increases, the static strength increases up to a specific hardness value (approximately 230 HB for the yield limit and 210 HB for the tensile strength), and subsequently decrease. The elongation and dynamic strength trendlines display a continuous decrease when the hardness increases.

References

1. Brezina, P. Heat treatment of complex aluminium bronzes. International Metals Reviews 1982, 27(2), 77–120. [Google Scholar] [CrossRef]
2. Vu, A.T.; Nguyen, D.N.; Pham, X.D.; Tran, D.H.; Vuong, V.H.; Pham, M.K. Influence of strengthening phases on the microstructure and mechanical properties of CuAl9Fe4 alloy. International Journal of Scientific & Engineering Research 2018, 9, 346–351. [Google Scholar]
3. Chau, M.Q.; Vu, A.T.; Le, T.S.; Mai, V.T.; Nguyen, D.N.; Doan, X.T.; Do, H.C.; Nguyen, D.T. Influence of tempering time on microstructure and mechanical properties of CuAl9Fe4 alloy. Journal of Mechanical Engineering Research and Developments 2021, 44(7), 75–85. [Google Scholar]
4. Slama, P.; Dlouhy, J.; Kövér, M. Influence of heat treatment on the microstructure and mechanical properties of aluminium bronze. Materials and Technology 2014, 48(4), 599–604. [Google Scholar]
5. Jain, P.; Nigam, P.K. Influence of heat treatment on microstructure and hardness of nickel aluminium bronze (Cu-10Al-5Ni-5Fe). Journal of Mechanical and Civil Engineering 2013, 4(6), 16–21. [Google Scholar] [CrossRef]
6. Aaltonen, P.; Klemetti, K.; Hannien, H. Effect of tempering on corrosion and mechanical properties of cast aluminium bronzes. Scandinavian Journal of Metallurgy 1985, 14, 233–242. [Google Scholar]
7. Mi, G.; Zhang, J.; Wang, H. The effect of ageing heat treatment on the mechanical properties of Cu-Al-Fe-(x) alloys. Key Engineering Materials 2011, 467-469, 257–262. [Google Scholar] [CrossRef]
8. Matijevic, B.; Sushma, T.S.K.; Prathvi, B.K. Effect of heat treatment parameters on the mechanical properties and microstructure of aluminium bronze. Technical Journal 2017, 11(3), 107–110. [Google Scholar]
9. Maximov, J.; Duncheva, G.; Anchev, A.; Dunchev, V.; Argirov, Y.; Todorov, V.; Mechkarova, T. Effects of heat treatment and severe surface plastic deformation on mechanical characteristics, fatigue and wear of Cu-10Al-5Fe bronze. Materials 2022, 15, 8905. [Google Scholar] [CrossRef] [PubMed]
10. Vuchkov IN, Vuchkov II. QStatLab Professional, v. 5.5 – statistical quality control software. User’s Manual, Sofia, 2009.
11. Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 2002, 6(2), 182–197. [Google Scholar] [CrossRef]