Experimental and Statistical Analysis of the Effect of Heat Treatment on Surface Roughness and Mechanical Properties of Thin-Walled Samples Obtained by Selective Laser Melting from the Material AlSi10Mg

Statistical analysis of mechanical properties of thin-walled samples (~500 microns) obtained by selective laser melting from AlSi10Mg material and subjected to heat treatment for 1 h at temperatures from 260 °C to 440 °C (step of aging temperature change 30 °C) has shown that the maximum strain hardening in the stretching diagram section from yield strength to tensile strength is achieved at the heat treatment temperature equal to 290 °C. At carrying out of correlation analysis, a statistically significant positive correlation between deformation corresponding to yield strength and the sum of heights of the largest protrusions and depths of the largest depressions of the surface roughness profile within the basic length of the sample (Rz) and the full height of the surface roughness profile (Rmax) was established. It was found that the reason for the correlation is the presence of cohesive states between the extreme values of the surface roughness profile that persist along the entire length of the specimen.


Introduction
One of the main directions of the development of modern industrial technologies is the creation of high-quality products with low production costs.Reduction of production costs can be achieved by reducing to the minimum possible time of creation of the final product-"from idea to finished product" with simultaneous preservation of high-quality manufacturing.Among the technologies actively introduced in the production process, additive manufacturing technologies fall under these requirements.
ISO/ASTM 52900:2015 classifies the technologies used in additive manufacturing and considers the type of raw materials, deposition techniques, and methods of melting or curing the material [1].The most common technologies of additive manufacturing are SLA and FDM printing [2][3][4][5][6]; these technologies use thermoplastics and polymer resins as the main materials, which limits the scope of application of products made by these technologies.Technologies that allow manufacturing products from metal, such as selective laser melting (SLM) technology [7][8][9][10][11][12][13][14][15], have a wider industrial potential.Powders of metals and alloys of various compositions are used as a starting material to produce final products using selective laser melting technology.
One of the most promising, from the industrial and environmental [16] point of view, materials for use in additive manufacturing are aluminum and aluminum-based alloys.The combination of low cost of aluminum and aluminum-based alloys, good mechanical properties, and additive technologies allows for a shortened production cycle of the final product and reduces the cost of the final product.
Currently, the list of aluminum-based alloys used in the manufacture of products using additive technologies, including methods of selective laser melting, is quite extensive [17,18].However, despite a large amount of research, there are still several problems that require consideration.Optimization of additive manufacturing parameters to obtain a product with specified structural properties, namely mechanical strength and surface roughness [17].
In the presented work, samples made by SLM technology from light alloy AlSi10Mg were studied.
AlSi10Mg has good mechanical strength and corrosion resistance [19][20][21][22] and allows the manufacture of products using SLM technology of complex geometric shapes [23].Kamarudin et al. [23] note that during the manufacture of complex-shaped products (molds), inhomogeneity of surface roughness and deviation of actual dimensions from the design dimensions are observed, which is attributed to the influence of local heat transfer.Studies [24,25] show that the effect of local heat transfer affects the microstructure of the bulk product and, consequently, the mechanical properties.In addition, the change in mechanical properties of the final product depends on the tilt angle of the product during printing.Changing the tilt angle from 35.5 • to 90 • leads to an increase in mechanical properties by 12% (as the angle increases) while the surface roughness decreases [24].
An increase of mechanical properties at the manufacturing of specimens by SLM technology from AlSi10Mg material is achieved due to hardening.The main mechanism of hardening is precipitate hardening, which contributes more than the hardening of Si solid solution in the α-Al matrix [25].Clarification of the mechanisms of mechanical properties enhancement of AlSi10Mg samples obtained by selective laser melting shows that precipitate strengthening is achieved due to a very thin Al-Si eutectic structure between α-Al dendrites and the formation of a microstructure oriented transversely to the direction of load application, and the anisotropy of properties becomes minimal when the scanning speed is optimized [26][27][28][29].Additional sources of improvement of mechanical properties of the samples are changes in the gas atmosphere in which selective laser melting is performed, changes in surface roughness and porosity, and thermal post-treatment of parts manufactured by SLM printing technology from AlSi10Mg.When argon was replaced by nitrogen in the working chamber of the SLM 3D printer, the achievement of the strength limit of ~350 MPa was recorded [29].
The influence of porosity and surface roughness of the samples obtained by SLM printing technology from AlSi10Mg has received a great deal of attention [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47].The focus of the works is related to the optimization of technological parameters to reduce surface roughness and porosity and, consequently, to increase hardness, impact toughness, and fatigue strength by reducing the surface roughness of samples obtained by SLM printing technology from AlSi10Mg.In particular, the critical point of energy density, which gives the minimum pore fraction for AlSi10Mg and is about 60 J/m 3 [41] and exposure time of 140-160 µs [42], was shown to exist.In addition to the optimization of technological parameters, the influence of different surface post-treatment methods on the mechanical properties of samples has been investigated [33,46].It is noted in [33] that strong vibration hardening had the greatest effect on the improvement of fatigue life, followed by laser hardening and shot peening.
However, the works do not analyze the changes in tensile mechanical properties as a function of surface roughness on thin-walled samples, where the contribution of the surface to the tensile strength may be significant.
The influence of thermal post-treatment on the mechanical properties of samples produced by SLM technology is under active study [48] and requires detailed elaboration.
In the works [31,46], the application of standard heat treatment T6 is considered, and it is shown that the average surface roughness of samples obtained by SLM technology from AlSi10Mg material decreased after heat treatment at 540 • C for 2 h.However, after artificial aging at 155 • C for 12 h and initially at 530 • C for 2 h, the surface roughness increased [34].The lack of significant hardening of the material under the standard T6 heat treatment regime is also confirmed [49].In [50], the occurrence of anisotropy of mechanical properties arising in horizontally annealed samples during heat treatment carried out at 270 • C for 1.5 h was demonstrated, and a decrease in properties compared to non-annealed samples was observed, indicating the need for further search for an optimal heat treatment regime.
The analysis shows that earlier, the connection between mechanical properties in the tension of thin-walled samples made by technology of selective laser melting of AlSi10Mg material (changes in the height of the surface profile is up to 40% of the thickness of the sample) and surface roughness has not been studied, as well as the question of choosing the optimum mode of heat treatment of thin-walled samples made by technology SLM of AlSi10Mg remains relevant.
Thus, the purpose of the presented work is to determine the effect of heat treatment at temperatures from 260 • C to 440 • C for 1 h on the tensile mechanical properties and surface roughness of thin-walled samples (~500 µm) manufactured by SLM technology from AlSi10Mg.

Mechanical Properties
The microstructure and chemical composition of the studied materials were analyzed using a Phenom ProX scanning electron microscope (Holland) equipped with an adapter for elemental analysis by energy dispersive spectroscopy.The surface roughness of the samples was measured using a HOMMEL-ETAMIC T8000 profilograph (JENOPTIK (Hommel-Etamic), Jena, Germany), mechanical tensile tests were performed on an INSTRON 5989 electromechanical testing machine (Instron, Norwood, MA, USA) at a speed of 2 mm/min and statistical analysis of the experimental results was performed using software (Rstudio 2023.06.1 Posit Software, PBC, GNU license) written in R language.

Production of Samples
AlSi10Mg powder served as a starting material for the fabrication of samples by selective laser melting.The size of the powders ranged from 30 µm to 75 µm.Figure 1 shows a micrograph of the starting material and the size distribution of the powder particles.
The average chemical composition of the initial AlSi10Mg powder is presented in Table 1.
Printing was carried out on a Farsoon FS121M SLM selective laser melting machine (Farsoon Technologies, Changsha, China) with a pre-installed laser with a maximum power of 500 W. The main printing modes were layer thickness 30 µm, laser power P = 340 W, hatching distance 0.15 mm, and laser travel speed 1500 mm/s.Figure 2 shows a schematic drawing of the sample and its location on the table during fabrication by selective laser melting.The influence of thermal post-treatment on the mechanical properties of samples produced by SLM technology is under active study [48] and requires detailed elaboration.In the works [31,46], the application of standard heat treatment T6 is considered, and it is shown that the average surface roughness of samples obtained by SLM technology from AlSi10Mg material decreased after heat treatment at 540 °C for 2 h.However, after artificial aging at 155 °C for 12 h and initially at 530 °C for 2 h, the surface roughness increased [34].The lack of significant hardening of the material under the standard T6 heat treatment regime is also confirmed [49].In [50], the occurrence of anisotropy of mechanical properties arising in horizontally annealed samples during heat treatment carried out at 270 °C for 1.5 h was demonstrated, and a decrease in properties compared to non-annealed samples was observed, indicating the need for further search for an optimal heat treatment regime.
The analysis shows that earlier, the connection between mechanical properties in the tension of thin-walled samples made by technology of selective laser melting of AlSi10Mg material (changes in the height of the surface profile is up to 40% of the thickness of the sample) and surface roughness has not been studied, as well as the question of choosing the optimum mode of heat treatment of thin-walled samples made by technology SLM of AlSi10Mg remains relevant.
Thus, the purpose of the presented work is to determine the effect of heat treatment at temperatures from 260 °C to 440 °C for 1 h on the tensile mechanical properties and surface roughness of thin-walled samples (~500 μm) manufactured by SLM technology from AlSi10Mg.

Mechanical Properties
The microstructure and chemical composition of the studied materials were analyzed using a Phenom ProX scanning electron microscope (Holland) equipped with an adapter for elemental analysis by energy dispersive spectroscopy.The surface roughness of the samples was measured using a HOMMEL-ETAMIC T8000 profilograph (JENOPTIK (Hommel-Etamic), Jena, Germany), mechanical tensile tests were performed on an IN-STRON 5989 electromechanical testing machine (Instron, Norwood, MA, USA) at a speed of 2 mm/min and statistical analysis of the experimental results was performed using software (Rstudio 2023.06.1 Posit Software, PBC, GNU license) written in R language.

Production of Samples
AlSi10Mg powder served as a starting material for the fabrication of samples by selective laser melting.The size of the powders ranged from 30 μm to 75 μm.Figure 1 shows a micrograph of the starting material and the size distribution of the powder particles.The average chemical composition of the initial AlSi10Mg powder is presented in Table 1.Printing was carried out on a Farsoon FS121M SLM selective laser melting machine (Farsoon Technologies, Changsha, China) with a pre-installed laser with a maximum power of 500 W. The main printing modes were layer thickness 30 μm, laser power P = 340 W, hatching distance 0.15 mm, and laser travel speed 1500 mm/s.Figure 2 shows a schematic drawing of the sample and its location on the table during fabrication by selective laser melting.AlSi10Mg samples made by selective laser melting technology were subjected to sandblast cleaning followed by heat treatment.The view of the samples after cleaning is shown in Figure 2C. Figure 3 shows the scanning strategy for the fabrication of samples using the SLM technique from AlSi10Mg.AlSi10Mg samples made by selective laser melting technology were subjected to sandblast cleaning followed by heat treatment.The view of the samples after cleaning is shown in Figure 2C. Figure 3 shows the scanning strategy for the fabrication of samples using the SLM technique from AlSi10Mg.
Heat treatment of the samples was carried out in a muffle furnace in a natural atmosphere at temperatures from 260 • C to 440 • C with a step of 30 • C. The samples were heated up to the holding temperature with the natural heating rate of the furnace, held for 1 h at the set temperature, and cooled down with the natural cooling rate of the furnace.The heat-treated specimens were subjected to surface roughness analysis and mechanical testing.The geometry of the specimens subjected to tensile testing is shown in Figure 2A.Heat treatment of the samples was carried out in a muffle furnace in a natural atmosphere at temperatures from 260 °C to 440 °C with a step of 30 °C.The samples were heated up to the holding temperature with the natural heating rate of the furnace, held for 1 h at the set temperature, and cooled down with the natural cooling rate of the furnace.The heat-treated specimens were subjected to surface roughness analysis and mechanical testing.The geometry of the specimens subjected to tensile testing is shown in Figure 2A.

Surface Roughness of Samples Manufactured by Selective Laser Melting Technology from AlSi10Mg Material
A total of 48 samples made by selective laser melting technology from AlSi10Mg material were subjected to a surface roughness study.
Figure 4 shows the results of surface roughness profile measurements for the samples that were not annealed after fabrication by selective laser melting technology.Similar roughness diagrams were obtained for the other 42 samples.
The analysis of the autocorrelation functions of the surface profile shows that there is a regularity in the variation of the surface profile height depending on the sample length, which has the character of a stationary series [50].
Table 2 presents the arithmetic mean values of the absolute values of profile deviations within the base length (Ra), the sum of the height of the largest profile protrusion, and the depth of the largest profile depression within the base length of the sample (Rz) and the total profile height (Rmax) of all samples.Table 3 presents the basic statistical analysis of the surface roughness parameters presented in Table 2.The analysis of basic statistical characteristics shows that the arithmetic mean of absolute values of profile deviations within the basic length (Ra) does not have a wide scatter for different samples.At the same time, the greatest profile height, the sum of the height of the greatest profile protrusion, and the depth of the greatest profile depression within the basic length of the sample (Rz) have rather high fluctuations of values from sample to sample, the same behavior is observed for the total profile height (Rmax).

Mechanical Test Results for Groups of Specimens Manufactured by Selective Laser Melting
Technology from AlSi10Mg Material, Pre-Treated at Different Temperatures A total of 48 specimens were subjected to tensile tests; 42 of them were pre-annealed at different temperatures.Figure 5 shows the engineering stress-strain curves of samples that did not undergo pre-annealing and samples that underwent pre-annealing at temperatures from 260

Mechanical Test Results for Groups of Specimens Manufactured by Selective Laser Melting Technology from AlSi10Mg Material, Pre-Treated at Different Temperatures
A total of 48 specimens were subjected to tensile tests; 42 of them were pre-annealed at different temperatures.Figure 5 shows the engineering stress-strain curves of samples that did not undergo pre-annealing and samples that underwent pre-annealing at temperatures from 260 °C to 440 °C.Preliminary analysis of the results of the basic statistical analysis shows that the maximum value of tensile strength and yield strength is achieved at annealing temperatures of 260 • C and 290 • C, while the maximum ductility is achieved at annealing temperature of 440 • C.

Statistical Analysis of the Results
The choice of the criterion for checking the experimental results belonging to the normal distribution is made based on calculations of the average statistical power depending on the number of tested samples.The average power of the criteria is calculated using the Monte Carlo method with the number of iterations equal to 100,000.During the calculations, a simple distribution was introduced into the criterion, the parameters of which were calculated by the likelihood method.The Cauchy, exponential, Gumbel, log-normal, logistic, normal, and Weibull distributions were considered simple distributions.The distribution parameters were iteratively recalculated depending on the number of tested samples.
Four criteria were selected for the study: • From parametric criteria: Shapiro-Wilk criteria [51]; D'Agostino criteria [52]; • From non-parametric: Kolmogorov-Smirnov criteria [53]; Anderson-Darling criteria [54]; Anderson-Darling criterion and D'Agostino criterion have limitations on the minimum number of studies; the number of studies must be greater than or equal to 7.
Figure 6 shows the results of calculating the average power of the statistical criterion depending on the number of trials.
Analysis of the results of calculations of the average statistical power of the four statistical criteria shows that the maximum power is possessed by the Kolmogorov-Smirnov criterion.The exception is the case when the measurement results obey the Cauchy distribution and the exponential distribution; when the number of trials is more than 40, the statistical power of the Anderson-Darling and Shapiro-Wilk criteria is almost equal to the power of the Kolmogorov-Smirnov criterion, and when the number of trials is more than 50, the power of the D'Agostino criterion approaches 1.For other distribution types, the statistical power of the Kolmogorov-Smirnov criterion is maximal.
To determine the theoretical distribution closest to the data, two information criteria were applied: Akaike and Bayesian.The results of applying the Akaike and Bayesian criteria are presented in Table 6.
Thus, the dependence of the average statistical power of the criterion on the number of studies is reflected in Figure 6G, and the lowest probability of making an error of the second kind when analyzing the experimental results presented in this paper occurs when using the Kolmogorov-Smirnov criterion.
Considering the results of modeling given in [55,56], the Kolmogorov-Smirnov criterion is the most applicable for data analysis in the problems of materials science, as it has the highest power and does not depend on the type of data distribution (in those cases when the closest type of data distribution are Weibull and Logistic distributions [56]).
Using the Kolmogorov-Smirnov criterion, the data in Tables 2 and 4 were tested for belonging to a normal distribution.Table 7 presents the results of the analysis.
Analysis of the results of applying the Kolmogorov-Smirnov test to surface roughness measurements and tensile test results show that the experimental values obtained do not belong to the normal distribution, and further statistical analysis should be carried out using non-parametric statistical criteria.
Of practical interest are the correlations between surface roughness parameters and mechanical properties, the change in mechanical properties of samples made by selective laser melting technology and annealing temperature, and the behavior of surface roughness as a function of sample length.At the first stage of the analysis, point diagrams of the dependence of mechanical properties on surface roughness parameters were plotted.Analysis of the results of applying the Kolmogorov-Smirnov test to surface roughness measurements and tensile test results show that the experimental values obtained do not belong to the normal distribution, and further statistical analysis should be carried out using non-parametric statistical criteria.
Of practical interest are the correlations between surface roughness parameters and mechanical properties, the change in mechanical properties of samples made by selective laser melting technology and annealing temperature, and the behavior of surface roughness as a function of sample length.At the first stage of the analysis, point diagrams of the dependence of mechanical properties on surface roughness parameters were plotted.Figure 7 shows an example of yield strength dependence on surface roughness parameters.Analysis of the graphs (Figure 7) shows that the yield strength of samples made by selective laser melting technology from AlSi10Mg material practically does not change Analysis of the graphs (Figure 7) shows that the yield strength of samples made by selective laser melting technology from AlSi10Mg material practically does not change depending on the main parameters of surface roughness and has a clearly expressed division of data into groups depending on the annealing temperature.
The behavior of the strength limit, strain corresponding to the yield strength, and strain corresponding to the strength limit depending on the main parameters characterizing the surface roughness did not show clearly expressed dependencies and stratification into groups.
The differences in the mechanical properties of the samples depending on the annealing temperature were analyzed using the Kruskal-Wallis criterion [57], the results of which are presented in Table 8.
The results of applying the Kruskal-Wallis criterion show that statistically significant differences are observed in the mechanical properties of samples obtained by selective laser melting technology annealed at different temperatures.No statistically significant differences were found in surface roughness parameters.A comparison of the test results (Table 8) with Figure 7 shows that mechanical properties do not have significant differences at all annealing temperatures.
To test pairwise differences between mechanical properties depending on annealing temperature, the Mann-Whitney test was applied [58].The results of the test are presented in Table 9.The results of applying the Mann-Whitney criterion show that statistically significant differences are observed at almost all combinations of annealing temperatures, and all considered mechanical properties, except for aging temperatures 260 • C and 290 • C differences in all mechanical properties are not statistically significant.Except for yield strength, the same situation is observed at aging temperatures 320 • C and 350 • C; strength, strain corresponding to yield strength and strain corresponding to tensile strength have no statistically significant differences.
Figure 8 shows the change in the average values of yield strength and tensile strength as a function of annealing temperature, strain corresponding to the tensile strength and yield strength, and the change in the strain hardening coefficient (θ = dσ dε ) [59] in the section of the tensile diagram from yield strength to tensile strength as a function of annealing temperature.
At an increase in aging temperature, there is a decrease in strength properties and an increase in the plasticity of samples obtained by selective laser melting technology from AlSi10Mg material (Figure 8A,B).
It follows from the presented dependences (Figure 8C) that the maximum strain hardening is achieved at the aging temperature equal to 290 • C. Considering the results of the analysis given in Table 9, the maximum strain hardening achieved is not statistically significantly different from the strain hardening achieved at 260 • C.
Figure 8 shows the change in the average values of yield strength and tensile strength as a function of annealing temperature, strain corresponding to the tensile strength and yield strength, and the change in the strain hardening coefficient ( ) [59] in the section of the tensile diagram from yield strength to tensile strength as a function of annealing temperature.At an increase in aging temperature, there is a decrease in strength properties and an increase in the plasticity of samples obtained by selective laser melting technology from AlSi10Mg material (Figure 8A,B).
It follows from the presented dependences (Figure 8C) that the maximum strain hardening is achieved at the aging temperature equal to 290 °C.Considering the results of the analysis given in Table 9, the maximum strain hardening achieved is not statistically significantly different from the strain hardening achieved at 260 °C.
The change of mechanical properties depending on the heat treatment temperature is associated with changes in the microstructure of the samples manufactured by SLM technology from AlSi10Mg material.Figure 9 shows the change in the microstructure of samples depending on the heat treatment temperature.The change of mechanical properties depending on the heat treatment temperature is associated with changes in the microstructure of the samples manufactured by SLM technology from AlSi10Mg material.Figure 9 shows the change in the microstructure of samples depending on the heat treatment temperature.
Metallographic analysis of the microstructure of the samples produced by SLM technology shows the presence of inhomogeneous microstructure across the width of the sample, which decreases with increasing heat treatment temperature.
Figure 8A shows the microstructure of AlSi10Mg alloy produced by SLM without heat treatment.Three different types of grain structure are observed in it.In the middle part of the melt pool, a fine grain structure is obtained.In the lateral parts of the melt pool, the grains are larger and elongated towards the heat source, which corresponds to that described in [45].A closer look at the microstructure of the samples without heat treatment on SEM shows a cellular microstructure [60], also having different cell sizes (Figure 8B).
Heat treatment of samples (Figure 9C) at 290 • C leads to a more uniform grain size distribution in the first and second zones from the melt center, while larger grains are retained at the melt boundary.Analysis of the microstructure of samples heat-treated at 290 • C with a scanning electron microscope (Figure 9D) shows a more uniform and less pronounced distribution of cells throughout the sample.With further heat treatment, the grain size becomes more uniform, and the cellular microstructure is no longer apparent (Figure 9E,F).When the heat treatment temperature is increased to 440 • C (Figure 9G), the microstructure becomes even more homogeneous, and inclusions of about 0.3 µm begin to appear on the samples.
Analysis of the microstructure of the cross-section of the sample shows the presence of grain structure (Figure 10B) and has a small number of inclusions (Figure 10A) similar in size to the inclusions presented in Figure 9H. Figure 10 shows the microstructure of the samples without heat treatment in crosssection with respect to the laser motion.
Heat treatment of samples (Figure 9C) at 290 °C leads to a more uniform grain size distribution in the first and second zones from the melt center, while larger grains are retained at the melt boundary.Analysis of the microstructure of samples heat-treated at 290 °C with a scanning electron microscope (Figure 9D) shows a more uniform and less pronounced distribution of cells throughout the sample.With further heat treatment, the grain size becomes more uniform, and the cellular microstructure is no longer apparent (Figure 9E,F).When the heat treatment temperature is increased to 440 °C (Figure 9G), the microstructure becomes even more homogeneous, and inclusions of about 0.3 μm begin to appear on the samples.
Figure 10 shows the microstructure of the samples without heat treatment in crosssection with respect to the laser motion.Analysis of the microstructure of the cross-section of the sample shows the presence of grain structure (Figure 10B) and has a small number of inclusions (Figure 10A) similar in size to the inclusions presented in Figure 9H.Thus, the maximum strain hardening of AlSi10Mg achieved at 290 • C can be attributed to the obtained sequential combination of fine and coarse-grained α-Al microstructure and the contribution of irregular eutectic phase varying with the heat treatment temperature [26][27][28][29].
To reveal not clearly expressed dependencies, correlation analysis of mechanical properties of samples obtained by selective laser melting technology from AlSi10Mg material and basic parameters describing surface roughness was applied.Considering the results of analyzing the distributions of the studied quantities (Tables 6 and 7, the distribution is different from normal), the correlation analysis by Kendall was applied.
Table 10 shows the Kendall correlation coefficients, the calculated level of statistical significance, and the coefficient of determination.The strength of the correlation was interpreted using the Evans scale.The level of statistical significance was assumed to be 0.05.The results of correlation analysis of mechanical properties of samples manufactured by selective laser melting technology from AlSi10Mg material and basic parameters of surface roughness show the presence of a weak statistically significant correlation between the strain corresponding to the yield strength and the sum of the height of the largest profile protrusion and the depth of the largest profile depression within the basic length of the sample (Rz) and between the strain corresponding to the yield strength and the full height of the profile (Rmax), in other cases statistically significant correlation between the strain corresponding to the yield strength and the full height of the profile (Rmax).
Figure 11 shows scatter diagrams of the dependences of the strain corresponding to the yield strength as a function of Rz and Rmax and regression models describing the established dependences.Table 11 presents the results of constructing the dependence of the strain corresponding to the yield strength on the surface roughness parameters.The obtained correlations and regression equations describe a statistically significant relationship between the experimentally obtained data but do not provide an answer to the causes of the found relationship.
To establish the reasons for the correlation relationship, the sum of the heights of the largest protrusions and depths of the largest depressions of the surface roughness profile within the base length of the sample (Rz) and the total height of the surface roughness profile (Rmax) were analyzed.
Rz is calculated by the equation: where  -height of the i-th protrusion of the surface roughness profile;  -depth of the i-th depression of the surface roughness profile.Rmax, respectively: where  -maximum height of roughness profile;  -maximum depth of surface roughness profile.
The analysis of the values included in Equations ( 1) and (2) shows that the main variables have extreme character, and their behavior should be analyzed by means of extreme value analysis [61].However, it should be taken into account that the correlation is observed with the value characterizing the sample as a whole, and the analysis should be performed based on the influence of extreme values on each other.
For these purposes, the autocorrelation function of the extreme values of the surface Table 11 presents the results of constructing the dependence of the strain corresponding to the yield strength on the surface roughness parameters.The obtained correlations and regression equations describe a statistically significant relationship between the experimentally obtained data but do not provide an answer to the causes of the found relationship.
To establish the reasons for the correlation relationship, the sum of the heights of the largest protrusions and depths of the largest depressions of the surface roughness profile within the base length of the sample (Rz) and the total height of the surface roughness profile (Rmax) were analyzed.
Rz is calculated by the equation: where y pmi -height of the i-th protrusion of the surface roughness profile; y vmi -depth of the i-th depression of the surface roughness profile.Rmax, respectively: where y max -maximum height of roughness profile; y min -maximum depth of surface roughness profile.The analysis of the values included in Equations ( 1) and (2) shows that the main variables have extreme character, and their behavior should be analyzed by means of extreme value analysis [61].However, it should be taken into account that the correlation is observed with the value characterizing the sample as a whole, and the analysis should be performed based on the influence of extreme values on each other.
For these purposes, the autocorrelation function of the extreme values of the surface roughness profile was analyzed.As a result of the analysis, it was found that statistically significant autocorrelation of maxima and minima is observed only for two samples: sample No. 3, aged at 380 • C, and sample No. 4, aged at 440 • C. Figure 12 shows the graphs of autocorrelation functions of maxima and minima for these samples.Removal of sample No. 3, aged at 380 °C, and sample No. 4, aged at 440 °C from the total sample, leads to the fact that the correlation between the strain corresponding to the yield strength and roughness parameters Rz and Rmax becomes statistically insignificant.Thus, the positive influence of surface roughness on the strain corresponding to the yield strength occurs when the maxima and minima of the surface roughness profile have a significant statistical correlation along the entire length of the sample.

Conclusions
As a result of statistical analysis of changes in mechanical properties and surface roughness depending on heat treatment, it was found that: 1. Maximum strain hardening of thin-walled samples made by selective laser melting technology from AlSi10Mg is achieved during the heat treatment for 1 h at 290 °C.2. The mechanical properties of AlSi10Mg samples are not statistically significantly different at 260 °C and 290 °C.3. The strain hardening of samples fabricated by SLM technology from AlSi10Mg is achieved due to the successive alternation of fine and coarse-grained α-Al microstructure and a more uniform distribution of the eutectic phase in the α-Al grain circle.4. At heat treatment of samples in the temperature range from 290 °С to 440 °С within one hour, there are no statistically significant changes in surface roughness.5.The correlation between the deformation corresponding to the yield strength and the sum of heights of the largest protrusions and depths of the largest depressions of the Removal of sample No. 3, aged at 380 • C, and sample No. 4, aged at 440 • C from the total sample, leads to the fact that the correlation between the strain corresponding to the yield strength and roughness parameters Rz and Rmax becomes statistically insignificant.Thus, the positive influence of surface roughness on the strain corresponding to the yield strength occurs when the maxima and minima of the surface roughness profile have a significant statistical correlation along the entire length of the sample.

Conclusions
As a result of statistical analysis of changes in mechanical properties and surface roughness depending on heat treatment, it was found that: 1.
Maximum strain hardening of thin-walled samples made by selective laser melting technology from AlSi10Mg is achieved during the heat treatment for 1 h at 290 • C.

2.
The mechanical properties of AlSi10Mg samples are not statistically significantly different at 260 • C and 290 • C.

3.
The strain hardening of samples fabricated by SLM technology from AlSi10Mg is achieved due to the successive alternation of fine and coarse-grained α-Al microstructure and a more uniform distribution of the eutectic phase in the α-Al grain circle.

4.
At heat treatment of samples in the temperature range from 290 • C to 440 • C within one hour, there are no statistically significant changes in surface roughness.

Figure 2 .
Figure 2. Schematic parameters of the sample made by selective laser melting method (A), its location on the table during printing (B), and samples made by selective laser melting method (C).

Figure 2 .
Figure 2. Schematic parameters of the sample made by selective laser melting method (A), its location on the table during printing (B), and samples made by selective laser melting method (C).

Materials 2023 , 24 Figure 3 .
Figure 3. Schematic representation of the scanning strategy for fabrication of samples by SLM technology from AlSi10Mg.

Figure 3 .
Figure 3. Schematic representation of the scanning strategy for fabrication of samples by SLM technology from AlSi10Mg.

Figure 6 .
Figure 6.Dependence of the average power of a statistical criterion as a function of the number of trials for four statistical criteria and seven different distributions.(A) Cauchy distribution; (B) exponential distribution; (C) Gumbel distribution; (D) log-normal distribution; (E) logistic distribution; (F) Normal distribution; (G) Weibull distribution.

Figure 6 .
Figure 6.Dependence of the average power of a statistical criterion as a function of the number of trials for four statistical criteria and seven different distributions.(A) Cauchy distribution; (B) exponential distribution; (C) Gumbel distribution; (D) log-normal distribution; (E) logistic distribution; (F) Normal distribution; (G) Weibull distribution.

Figure 7
Figure 7 shows an example of yield strength dependence on surface roughness parameters.

Figure 7 .
Figure 7. Dependence of yield strength on surface roughness parameters at different annealing temperatures of samples.(A) from Ra; (B) from Rz; (C) from Rmax.

Figure 7 .
Figure 7. Dependence of yield strength on surface roughness parameters at different annealing temperatures of samples.(A) from Ra; (B) from Rz; (C) from Rmax.

Figure 8 .
Figure 8. Dependence of average values of strength and yield strengths (A), strains corresponding to strength and yield strength (B), and strain hardening on the aging temperature of samples (C) obtained by selective laser melting technology from AlSi10Mg material.

Figure 8 .
Figure 8. Dependence of average values of strength and yield strengths (A), strains corresponding to strength and yield strength (B), and strain hardening on the aging temperature of samples (C) obtained by selective laser melting technology from AlSi10Mg material.

Figure 9 .
Figure 9. Microstructure of samples obtained by selective laser melting technology from AlSi10Mg material in the direction of laser motion and heat treated at (A) No heat treatment (Optical microscope); (B) No heat treatment (Scanning electron microscope); (C) heat treatment at 290 • C (Optical microscope); (D) heat treatment at 290 • C (Scanning Electron Microscope); (E) heat treatment at 380 • C (Optical Microscope); (F) heat treatment at 380 • C (Scanning Electron Microscope); (G) heat treatment at 440 • C (Optical Microscope); (H) heat treatment at 440 • C (Scanning Electron Microscope).(The smooth, straight lines in the figures are scratches left after polishing the samples).

Figure 10 .
Figure 10.Microstructure of samples without heat treatment in cross-section with respect to laser motion.(A) Scanning electron microscope.(B) Optical microscope.

Figure 10 .
Figure 10.Microstructure of samples without heat treatment in cross-section with respect to laser motion.(A) Scanning electron microscope.(B) Optical microscope.

Materials 2023 , 24 Figure 11 .
Figure 11.Dependence of strain corresponding to yield strength on (A) Rz, (B) Rmax, and regression models describing the dependence of correlated values.

Figure 11 .
Figure 11.Dependence of strain corresponding to yield strength on (A) Rz, (B) Rmax, and regression models describing the dependence of correlated values.

Materials 2023 , 24 Figure 12 .
Figure 12.Autocorrelation functions of minima and maxima of the surface roughness profile.(A) ACF minima of sample No 4 aging at 440 °C; (B) ACF maxima of sample No 4 aging at 440 °C; (C) ACF minima of sample No 3 aging at 380 °C; (D) ACF minima of sample No 3 aging at 380 °C (The horizontal blue line indicates the level of statistical significance).

Figure 12 .
Figure 12.Autocorrelation functions of minima and maxima of the surface roughness profile.(A) ACF minima of sample No 4 aging at 440 • C; (B) ACF maxima of sample No 4 aging at 440 • C; (C) ACF minima of sample No 3 aging at 380 • C; (D) ACF minima of sample No 3 aging at 380 • C (The horizontal blue line indicates the level of statistical significance).

Table 1 .
Average chemical composition of AlSi10Mg powder.

Table 1 .
Average chemical composition of AlSi10Mg powder.

Table 2 .
Surface roughness of samples produced by selective laser melting technology from AlSi10Mg.

Table 3 .
Basic statistical analysis of the results of surface roughness measurements of 48 samples manufactured by selective laser melting technology from AlSi10Mg.

Table 4 presents
the main mechanical properties of 48 tested samples.

Table 4 presents
the main mechanical properties of 48 tested samples.

Table 4 .
Basic mechanical properties of samples obtained by selective laser melting from AlSi10Mg material.

Table 5
presents the basic statistical analysis of mechanical parameters presented in

Table 5 .
Basic statistical analysis of tensile test results of 48 specimens fabricated by selective laser melting technology from AlSi10Mg.

Table 6 .
Closest distribution types according to the minimum of Akaike and Bayesian criteria.

Table 6 .
Closest distribution types according to the minimum of Akaike and Bayesian criteria.

Table 7 .
Results of testing whether the data in Tables2 and 4belong to a normal distribution.

Table 8 .
Results of applying the Kruskal-Wallis criterion to the data given in Tables2 and 4in the study of the influence of annealing temperature on mechanical properties and surface roughness characteristics.

Table 9 .
Results of the analysis of statistical differences in groups of samples aged at different temperatures.

Table 10 .
Results of Kendall correlation analysis between the main mechanical properties and surface roughness parameters of the samples obtained by selective laser melting technology from AlSi10Mg material.

Table 11 .
Equations describing a weak correlation between the strain corresponding to yield strength and surface roughness parameters.

Table 11 .
Equations describing a weak correlation between the strain corresponding to yield strength and surface roughness parameters.