Introduction

Spheroidal graphite iron (SGI) is a multifaceted material within the cast iron grades and is used in various applications to substitute for steel. In order to fulfill the increasing requirements on the material and to comply with the more and more precise process specifications, it is essential to have a strict process and quality control during SGI production.1 In addition to monitoring the chemical composition, it is important to evaluate the nucleation and solidification sequence of the melt before as well as after magnesium treatment and inoculation in order to achieve optimal mechanical properties.2,3 The inoculation of the melt influences not only the formation of graphite but also the eutectic temperature and thus the precipitation of graphite during solidification. Moreover, the eutectoid temperature is also affected, which can significantly influence the pearlite formation and subsequently the mechanical properties.4,5 Such a process control requires extensive knowledge of monitoring by use of the thermal analysis (TA).

The quality control of cast iron melts in iron foundries is mainly carried out by thermal analysis and optical emission spectroscopy (OES). Thermal analysis is described as the recording of a temperature-time-curve during the solidification in a specially prepared measuring cup and the subsequent analysis of this curve.6,7 The characteristic curve of the thermal analysis provides information on various aspects of the solidification behavior, such as the graphite morphology or the carbon equivalent (CE), which allows the melt with several alloying elements to be treated like a binary Fe-C alloy. Therefore, thermal analysis provides important information which extends far beyond the chemical composition.7 In addition to the microstructural properties of cast iron, correlations of specific mechanical properties can also be described with the temperature-time curve of thermal analysis. However, these analysis possibilities have not been fully exploited and are not used commercially yet.8

Although the mechanical properties of cast iron are the most important property for commercial use, thermal analysis has not yet been sufficiently studied in terms of its predictive power over a wide range of mechanical parameters. Kano et al.8 were the first to attempt to predict the mechanical properties using TA data. For this purpose, they applied a thermocouple to each of three molds to find out the eutectic graphitization ability (EGA). As a result, a correlation between EGA and tensile strength (TS) was found out, which is shown in Eqn. 1. This eutectic graphitizability describes the ability of carbon to convert into graphite. It is influenced by various factors such as the chemical composition or the cooling rate. In general, the higher the EGA, the higher the graphite content in the material.9 Using a large amount of data, Glover et al.10 performed a regression analysis in 1982 to work out a correlation between UTS (in psi) and liquidus arrest (TLA in °C). Thereby, the liquidus arrest temperature is the range of thermal standstill during the phase transformation from the liquid to the solid state. The result of this analysis is shown in Eqn. 2 with an R2 of 0.55. This correlation coefficient is not surprising since major microstructural characteristics, such as graphite shape, number of dendrites, spacing of dendrite arms, and eutectic grain size, have a rather small effect on liquidus arrest but a large one on mechanical properties.

$$ {\text{UTS}} = \left( {180 \cdot {\text{EGA}} + 170} \right) \cdot \left( {4.4 - {\text{CE}}} \right) + 160 $$
(1)
$$ {\text{UTS}} = - 388.447 + 357 \cdot T_{{{\text{LA}}}} $$
(2)

To improve the correlation, Glover et al. introduced the dendrite interaction area into the analysis, which is a qualitative estimate of the area fraction of the dendrites. The new equation had a higher correlation coefficient with an R2 value of 0.72. The correlation is shown in Eqn. 3.

$$ {\text{UTS}} = - 248.504 + 234 \cdot T_{{{\text{LA}}}} + 93 $$
(3)

This result shows the potential that thermal analysis brings to the prediction of mechanical properties.7,8,10 With the many different characteristic temperatures, thermal analysis offers a large number of characteristics that have not yet been investigated in terms of their statistical correlation with various mechanical properties and characteristic values. As previously shown, by considering a single characteristic temperature (like TLA) alone, a high accuracy for the prediction of tensile strength could be found out. The relationship with other mechanical characteristic values was not investigated. In addition, the investigation of other characteristic points and their correlation with the mechanical parameters offers high potential. If several correlations are found, the predictive power can be increased by combining different explanatory characteristic points. In this context, the thermal analysis also allows a closer examination of the temporal component in addition to the temperature variables. If a prediction of the mechanical characteristic values can be determined only by thermal analysis, this would offer considerable potential for production, since time-consuming and expensive quality measurements could be substituted by an analysis of the cooling curve. This provides the basis for the research question of this thesis and the relationship between the characteristics of the thermal analysis and the mechanical properties of ductile cast iron will be investigated in more detail in the course of this thesis.

The aim of this work is to develop an analysis concept that makes it possible to analyze the time-temperature curves and store the characteristic data in a scalable database. In addition, a new evaluation method will be developed to increase the informative value of the thermal analysis to show a new way of predicting the material properties of SGI. Finally, the developed models are to be validated with specially performed castings in the foundry institute. The developed models should be able to predict the mechanical properties, independent of the wall thickness of the casting. For this purpose, the characteristic data of the thermal analysis in connection with the tensile strength, the elongation at fracture, the uniform elongation, the yield strength and the modulus of elasticity are to be investigated.

Design of Experiments

In order to be able to evaluate the data from the thermal analysis reliably, a suitable cooling curve analysis tool for determining the characteristic temperatures of the cooling curve was developed as part of this work using a spreadsheet calculation software.

Development of the Cooling Curve Analysis Tool

The temperature and time curves from the file of the thermal analysis are automatically read into the cooling curve analysis tool and the first derivative of the cooling curve is formed. This corresponds to the change in temperature over time, i.e., the cooling rate \(\dot{T}\). At a phase transition during solidification, there is a strong change in the cooling rate. By analyzing the change in cooling rate considering the temperature-time curves from TA, the characteristic temperatures with their respective points in time can be determined. Those characteristic points considered in the cooling rate analysis and the conditions established to be able to identify the points in the cooling rate, or cooling curve, are shown in Table 1.

Table 1 Determined Characteristic Temperatures and Solidification Times as Well as the Imposed Conditions.
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The different temperatures Tx describe the temperature that the melt has at the characteristic point of the transformation. The different times tx describe the duration from the beginning of the measurement until the respective characteristic point x is reached. After the points have been recognized by the analysis tool, the position of the points is automatically plotted in the course of the cooling rate as shown in Figure 1.

Figure 1
figure 1

Graphical representation of the cooling curve (blue) and the cooling rate (orange), as well as highlighting the characteristic points.

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Data Collection and Generation

The cooling curves of the thermal analyses used for this work are all originate from various castings at the Foundry Institute of RWTH Aachen University. A total cooling curves from the last 10 years were analyzed and, if available, the associated mechanical properties were documented. In addition to the characteristic temperatures and solidification times, special attention is paid to the temperature differences and the cooling duration as well as the cooling rate during data collection, since these characteristics can have an influence on the graphite formation and the pearlite and ferrite content in the metal matrix and thus significantly affect the mechanical properties. Table 2 shows all the data points considered.

Table 2 Calculated Data Using the Characteristic Points.
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All data points of the temperature difference and time difference were automatically calculated and saved by the analysis tool. For the calculation of the temperature difference \(\Delta T\) and the time difference \(\Delta t\), the distances of the temperatures \(T_{x}\) and \(T_{y}\) to be considered at the respective times \(t_{x}\) and \(t_{y}\) are taken. The calculation of the cooling rates was performed in a similar way, so that the cooling rate between two characteristic points is determined. In addition to the data points listed in Table 2, recalescence and undercooling are calculated for each cooling curve in the thermal analysis.

The mechanical property for which the relation with each data point was investigated is the ultimate tensile strength UTS. Where available, for the geometries of a YII wedge and YIV wedge according to the European standard DIN EN 156311 as well as the geometry of a cylinder, these were added manually from the results of the different castings to the corresponding results of the collected data of the respective thermal analysis. The different geometries are presented in more detail in the following section.

Data Analysis

For the analysis of the data, the procedure of supervised learning was used. Linear regression analysis was selected as the analysis algorithm because it has already been shown to be a suitable analysis algorithm in some areas of foundry engineering in connection with thermal analysis, and regression analysis is also frequently used in the data science field to investigate initial correlations.7,12,13,14

Using a spreadsheet software, a linear regression analysis was performed with all collected data points and the different mechanical parameters of the different geometries. In this analysis, the collected data points were used as predictor variables to examine the mechanical characteristic values. The results were then compared. The coefficient of determination R2 was used as a comparison value and for a general evaluation of the correlation of the predictor variable with the mechanical properties. Depending on the field of application, a coefficient of determination of 0.2 is titled as sufficient correlation. Therefore, variables with a coefficient of determination greater than 0.2 were filtered out for the construction of a final computational model. In addition, it is assumed that the mechanical properties of cast iron could not be predicted by only one variable, but very likely depended on several variables. Since the correlation of a data point with the mechanical properties of all three geometries must be considered and a different amount of data is available for the mechanical properties of the different geometries, special rule was established for the selection of the final variables. This rule is as follows: If a coefficient of determination greater than 0.2 is observed for a variable used to predict the mechanical properties of the cylinder, a correlation close to 0.2 must also be observable for the other geometries, the YII wedge and YIV wedge.14,15

Experimental Procedure

A casting was made to validate the models created. For casting, two identical molds were built, each with a YII, a YIV and a cylinder geometry. One mold for taking samples for mechanical testing and the other for recording the temperature over time for the different geometries. The casting system is shown in Figure 2.

Figure 2
figure 2

Casting system with positioning of the thermocouples.

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After the materials have been melted in the furnace and heated to a temperature of 1500 °C, the melt is poured into a preheated ladle. There, the cover scrap, magnesium and inoculant are added. Before casting occurred, a small amount of the melt was removed to perform a thermal analysis. The melt removed from the ladle is poured into an empty QuiK-Cup as well as a QuiK-Cup filled with tellurium and sulfur. The QuiK-Cups are equipped with thermocouples. This allows the temperature profile during cooling to be recorded. The characteristic points of the thermal analysis of the stably solidified QuiK-Cup are used to predict the mechanical parameters of the different geometries of the cast.

After the melt was taken for thermal analysis, the casting of the test specimens occurred. The melt was successively poured into the two prepared molds. One of the molds was prepared with thermocouples before casting in order to record the temperature profile of the different geometries. The thermocouples were placed in the casting at the points where the specimens for mechanical testing were later taken. The position of the thermocouples is shown schematically in Figure 2. For each casting, tensile specimens were made at the locations of the thermocouples marked in Figure 3.

Figure 3
figure 3

Schematic diagram of the position of the thermocouples in the YII and YIV wedge, as well as in the cylinder.

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In order to be able to apply the model for different geometries, which should be done before with the help of a conversion factor, the thermal module was included in the regression analysis. The modulus describes the ratio of volume and surface area of the casting. Table 3 shows the moduli for all three geometries.

Table 3 Thermal Modules of the Different Geometries.
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By using the thermal module, with a listing of all the descriptive variables, with their associated modules and mechanical characteristic values, models could be created to predict the different mechanical characteristic values. The characteristic points of the QuiK-Cup measured during the thermal analysis of the casting are then inserted into the models created and compared with the mean values of the mechanical characteristic values actually measured.

Results

For the linear regression analysis of the tensile strength using the created model, a total of 110 tensile specimens were considered and related to the characteristic data points from the respective thermal analyses.

Characteristic Temperatures and Times

The results of the regression analyses of the characteristic temperatures and the corresponding solidification times are documented in Table 4 by the respective coefficient of determination. The graphs of the best results of the YII and YIV wedge and for the cylinder are shown in Figure 4.

Table 4 Measures of Determination of Characteristic Temperatures for Tensile Strength.
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Figure 4
figure 4

(a) Plots of the best results of linear regression analysis for tensile strength versus eutectoid temperature \({T}_{ETo}\) for cylinder geometry. (b) Graphs of the linear regression analysis for tensile strength above the duration until eutectoid solidification \({t}_{{\text{ETo}}}\) for the cylindrical castings.

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The cells highlighted in bold mark the relevant points that exceed a coefficient of determination of 0.2. The unmarked cells are below the threshold of 0.2.

As can be seen from the results, both \(T_{{{\text{liq}}}}\) and \(t_{{{\text{liq}}}}\) are not suitable to describe the tensile strength, respectively, have a small influence on the strength. Also, the start (\(T_{{\text{eut, start}}}\)/\(t_{{\text{eut, start}}}\)) as well as the end of eutectic solidification (\(T_{{\text{eut,end}}}\)/\(t_{{\text{eut, end}}}\)) show only a low explanatory correlation for the tensile strength both for the characteristic temperatures and the times. The strongest correlations were observed for the YII wedge and the YIV wedge in the region of the eutectic point. Here, the coefficient of determination stays within a range of 0.5. \(T_{{\text{eut, min}}}\), \(T_{{{\text{eut}}, \max }}\) and \(T_{{{\text{eut}}}}\) thus show an explanatory relationship for the tensile strength for the YII wedge and the YIV wedge.

For the cylinder as well as for the YII and YIV wedge, the tensile strength increases with a higher \(T_{{{\text{ETo}}}}\) (see Figure 4a). With respect to the solidification times of the characteristic temperatures, there is only one value that exceeds the 0.2 limit; the duration until eutectoid solidification \(t_{{{\text{ETo}}}}\). This is the case only for the cylinder geometry. A negative influence can be observed here, as can be seen in Figure 4b. The longer the duration until the eutectoid transformation during the thermal analysis, the lower values result for the tensile strength of the cylinder and the other geometries. However, since the values for the other specimens break the special rule, no characteristic time points are considered in the model.

Temperature Difference and Time Differences

The results of the simple linear regression analysis for the previous presented temperature differences are shown in Table 5.

Table 5 Measures of Determination of the Difference in Characteristic Temperatures for Tensile Strength.
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When observing the temperature differences, there are only a few conspicuous features. In the eutectic solidification range, i.e., for \(\Delta T_{{\text{eut, start}}} - T_{{\text{eut, min}}}\), \(\Delta T_{{\text{eut, min}}} - T_{{{\text{eut}}}}\), \(\Delta T_{{{\text{eut}}}} - T_{{\text{eut, max}}}\), and \(\Delta T_{{\text{eut, start}}} - T_{{\text{eut, end}}}\), no correlation was found for any geometry. Recalescence and undercooling also showed no significant correlation. In the transition of the eutectic temperature to the eutectoid temperature, the highest coefficients of determination were shown. Both \(\Delta T_{{\text{eut,end}}} - T_{{{\text{ETo}}}}\) and for \(\Delta T_{{{\text{eut}}}} - T_{{{\text{ETo}}}}\), have a comparatively high R2 value with 0.62 and 0.50 for the cylinder, respectively. However, it is noticeable that in the case of \(\Delta T_{{{\text{eut}}}} - T_{{{\text{ETo}}}}\), the YII¬ and YIV values are very low, whereas for \(\Delta T_{{\text{eut,end}}} - T_{{{\text{ETo}}}}\), these show at least a minimal correlation with 0.22 and 0.20, respectively. Therefore, due to the special rule established in previously, only \(\Delta T_{{\text{eut,end}}} - T_{{{\text{ETo}}}}\) is included in the final model. As can be seen in Figure 5, there is a negative correlation between the magnitude of the temperature difference and the tensile strength. Thus, a larger temperature window appears to be associated with a lower tensile strength.

Figure 5
figure 5

Graphs of the correlations of the temperature difference between the end of eutectic solidification and eutectoid solidification \(\Delta {T}_{{\text{eut}},{\text{end}}}-{{\text{T}}}_{{\text{ETo}}}\); left: YII wedge; middle: YIV wedge; right: cylinder.

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The coefficients of determination of the temporal distance \(\Delta t\) are shown in Table 6. More correlations can be seen here than for the temperature difference. Again, the correlations are much stronger in the region between the eutectic temperature and the eutectoid temperature. The time intervals from the liquidus temperature to the end of eutectic solidification do not reach an R2 value of more than 0.2 a single time. The duration between the end of eutectic solidification to reaching the eutectic point, on the other hand, seem to be able to serve as an explanatory variable. Both \(\Delta t_{{\text{eut,end}}} - t_{{{\text{ETo}}}}\) and \(\Delta t_{{{\text{eut}}}} - t_{{{\text{ETo}}}}\) for the cylinder, with R2 values of 0.44 and 0.54, respectively, exceed the 0.2 limit. In this regard, it should be mentioned that the coefficients of determination for \(\Delta t_{{\text{eut,end}}} - t_{{{\text{ETo}}}}\) and \(\Delta t_{{{\text{eut}}}} - t_{{{\text{ETo}}}}\) exceed 0.2 for all geometries, whereas for \(\Delta t_{{{\text{liq}}}} - t_{{{\text{ETo}}}}\) only the cylinder does. The results for this cooling time from the liquidus temperature to the eutectoid point \(\Delta t_{{{\text{liq}}}} - t_{{{\text{ETo}}}}\), with 0.18 for YII, 0.20 for YIV and 0.47 for the cylinder, are striking. Some graphs for the three variables where an explanatory relationship for tensile strength was found are shown as examples in Figure 6. As can be seen, a negative trend can be observed for all values. For all three data points, \(\Delta t_{eut} - t_{ETo}\), \(\Delta t_{eut,end} - t_{ETo}\) and \(\Delta t_{liq} - t_{ETo}\), the tensile strength decreases with increasing solidification time between the points.

Table 6 Measures of Determination of the Time Intervals Between the Characteristic Temperatures for Tensile Strength.
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Figure 6
figure 6

Regression analyses of the best results for the time differences of the YIV wedge.

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Cooling Rates

The final results for tensile strength are those for cooling rate. The results for \(\dot{T}\) are listed in Table 7.

Table 7 Measures of Determination of the Cooling Rates for Tensile Strength.
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The results are very similar to the results for the time differences. Again, the strongest correlations are observed in the range between the eutectic temperature and the eutectoid temperature at \(\dot{T}\)eut,end − ETo and \(\dot{T}\)eut − ETo. The cooling rate over the entire solidification \(\dot{T}\) liq − ETo also shows an explanatory relationship. For \(\dot{T}\) eut – ETo and \(\dot{T}\)liq – ETo, each R2 value exceeds the 0.2. For \(\dot{T}\) eut,end–ETo, only the correlation for YII and YIV reaches an explanatory correlation between the thermal analysis values and the tensile strength with an minimum R2 of 0.2. Since the value of \(\dot{T}\)liq−ETo is very close to 0.2 and shows a correlation, this variable is included in the final model. Figure 7 shows the graphs of the YIV wedges for all three salient data points. All three data points show a positive correlation between increasing cooling rate and tensile strength.

Figure 7
figure 7

Relationship between cooling rate of the thermal analysis with the tensile strength of the YIV wedge.

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Modeling

The data points listed in Table 8 were used for model development because a coefficient of determination greater than 0.2 was observed in each of these data points for at least one geometry and they did not violate the special rule. As already described, the thermal modulus was used to connect the results of the different geometries. The explanatory relationship of the modulus with the tensile strength is shown in Figure 8.

Table 8 Explanatory Variables for the Tensile Strength Calculation Model.
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Figure 8
figure 8

Explanatory relationship between modulus and tensile strength.

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As the thermal modulus increases, the tensile strength decreases. This fits with the results of the time variable, since a larger modulus is associated with a slower cooling rate. However, the thermal modulus only shows a coefficient of determination of 0.13 and thus does not exceed the value of 0.2. Since the thermal modulus influences the cooling rate and thus also the cooling time, and is primarily used to show a transferability of the results of the thermal analysis to other components, an exception to the rule is made here. The rather weak correlation should be kept in mind for the calculated results of the model.

The coefficients of the descriptive variables determined with the multiple regression analysis are listed in Table 9. The tensile strength model calculated using spreadsheet calculation software also has a correction factor UTS of − 14612.5. The coefficient of determination of the model is 0.77.

Table 9 Coefficients of the Model for Calculating Ultimate Tensile Strength.
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Validation by Casting Tests

Using multiple regression analysis and the modulus of the different wall thicknesses, it was possible to develop a model that represents the tensile strength as a function of all previously observed explanatory variables. With the help of the model, which is shown in Eqn. 4, the tensile strength for the YII wedge and YIV wedge, as well as for the cylinder, could be calculated by inserting the characteristic data points.

$$ \begin{aligned} {\text{UTS}} = & 11.38 \cdot T_{TS1} - 19.24 \cdot T_{TS2} + 13.8 \cdot T_{TS3} + 6.2 \cdot T_{TS4} - 0.07 \cdot \Delta T_{TS1} - \Delta t_{TS1} + 3.94 \cdot \Delta t_{TS2} \\ & \quad - 749.22 \cdot \dot{T}_{TS1} + 2467.98 \cdot \dot{T}_{TS2} + 267.98 \cdot \dot{T}_{TS3} - 21.09 \cdot M_{TS} - 14612.5 \\ \end{aligned} $$
(4)

In order to be able to evaluate the results of the calculated mechanical properties, they were compared with the values of the mechanical test. For this purpose, the tensile strength was determined for all three geometries and compared with the calculated values. The results of the values calculated with the aid of the model and the measured mechanical values and the percentage deviation are shown in Table 10.

Table 10 Results of the Calculation and the Mechanical Test.
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The model predicts a decrease in tensile strength with increasing wall thickness. This was also found in the values of the mechanical test. In addition, the deviations are only very small. The YII wedge and the YIV wedge show only a small deviation of 6.9% and 8.31%, respectively. The cylinder even shows a deviation of only 2.32% here. In addition, it can be observed that the model estimates the tensile strength of the respective geometries to be higher than they actually are.

Discussion

In the following, we will first discuss the significance of the data examined. Then, the correlations found will be discussed in more detail, and finally, the models developed in the course of this work will be critically examined.

Significance of the Examined Data

For the prediction of the mechanical characteristic values of a wide variety of geometries, the data of the thermal analysis of a QuiK-Cup were analyzed and correlated with the mechanical properties of other geometries. When analyzing the data and the respective correlation with the mechanical characteristic values, it is first questionable whether the data points of the thermal analysis of the QuiK-Cup geometry can make a statement about other geometries, since two important assumptions were made for this work:

  1. 1.

    The characteristic temperatures of the thermal analysis of the QuiK-Cup are identical to the characteristic temperatures of other wall thicknesses.

  2. 2.

    If the thermal analysis of the QuiK-Cup of a grade "A" shows a faster cooling rate than the thermal analysis of the QuiK-Cup with grade "B", the cooling rate of another wall thickness is also faster for "A" than for "B

Influence of Characteristic Temperatures and Temperature Differences

In this work, the characteristic temperatures of the thermal analysis as well as the temperature differences between the characteristic temperatures were investigated for the relationship with various mechanical properties of ductile cast iron.

For the characteristic temperatures, the results for \(T_{{\text{eut,min}}}\), \(T_{{{\text{eut}}}}\), \(T_{{\text{eut,max}}}\) and \(T_{{{\text{ETo}}}}\) were the most striking. The temperatures in the eutectic solidification range have shown an explanatory relationship for the tensile strength. Here, an increase in the three temperatures in the eutectic range leads to an increase in tensile strength. In the eutectic temperature range, it is mainly the graphite spheres which influence the mechanical properties, which is why the question of the relationship between the eutectic temperatures and the formation of the graphite plays a decisive role here.

There are two main reasons that lead to an increase in the eutectic temperature. One is due to a longer solidification time and the other is due to alloying elements that influence the eutectic temperature. An increase in the eutectic temperature due to the longer solidification time would be accompanied by a lower nodularity and growth of the graphite spheres.16,17 However, these two factors would lead to a decrease in tensile strength, which does not fit with the results of the model, but with the results of the mechanical investigation, since the decrease in the cooling rate and the accompanying described graphite characteristics are reasons for lower tensile strengths of the YIV wedges and cylinders compared to the YII wedge. The slope of the eutectic temperature here is most likely more related to the alloying of the melt and thus to the chemical composition.

Furthermore, when looking at the data, it is questionable whether the minimum and maximum eutectic temperature here really means a strong correlation with the mechanical properties. Figure 9 shows the comparison of the eutectic temperature \(T_{{{\text{eut}}}}\) with the minimum \(T_{{\text{eut,min}}}\) and maximum eutectic temperature \(T_{{\text{eut,max}}}\). It can be seen that the eutectic maximum and eutectic minimum are very close to the eutectic temperature for almost every data point. This means the relationship of \(T_{{\text{eut,min}}}\) and \(T_{{\text{eut,max}}}\) with the mechanical properties is explained by the close location to the eutectic point and the associated properties. The coefficients of determination of the eutectic temperatures also speak for this: for each mechanical property, the coefficient of determination of \(T_{{{\text{eut}}}}\) is the highest.

Figure 9
figure 9

Correlation of eutectic temperature with eutectic maximum (left) and minimum (right) temperature.

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The temperature difference was analyzed mainly because the subcooling has an influence on the microstructure formation. However, the undercooling did not show any influence during the investigation. To a high degree of probability, this is related to the only minimal difference of \(T_{{\text{eut,min}}}\) and \(T_{{{\text{eut}}}}\) shown by the data, which in turn could be related to a good inoculation of the studied thermal analyses. Thus, the influence of supercooling could not be shown with the available data.

A correlation was observed for the temperature differences from the end of eutectic solidification to the eutectoid point. However, since the correlation is only very small in all observed cases, the correlation of this temperature difference is very likely rather due to the influence of the eutectoid temperature. This is also supported by the fact that a smaller temperature difference associated with an increasing eutectoid temperature.

The results have shown that both the chemical composition and the cooling rate have an overriding influence on the characteristic temperatures. It is very likely that a combination of both factors explains the observed results, which is not demonstrated due to the chemical compositions not considered in this work.

Influence of Cooling Times and Cooling Speeds

In addition to the characteristic temperatures and the associated temperature difference, special attention was paid in this work to the temporal component of the characteristic temperatures of the thermal analysis and what relationship they show with the mechanical properties. The temporal difference was studied to further investigate the influence of the temporal spacing of the characteristic points during solidification. Since solidification is driven by a temperature loss over time, the cooling rate is another factor that was investigated, since the literature shows significant correlations of the cooling rate and the microstructure formation and thus with the mechanical properties.

For all investigated specimen geometries, correlations with the mechanical properties were found. Thus, the cooling time \(t_{ETo}\) from the beginning of the data measurement until the eutectoid temperature was reached showed a correlation for the tensile strength. The same is true for the temperature difference \(\Delta t_{{\text{eut,end}}} - t_{{{\text{ETo}}}}\), \(\Delta t_{{{\text{eut}}}} - t_{{{\text{ETo}}}}\), \(\Delta t_{{{\text{liq}}}} - t_{{{\text{ETo}}}}\) and the cooling rates \(\dot{T}\)eut,end−ETo, \(\dot{T}\)eut−ETo and \(\dot{T}\)liq−ETo. The results have shown that for the data collected in this work, a longer cooling time to the eutectoid point is associated with poorer tensile strength. The results of solidification times and cooling rates from the end of eutectic solidification to the eutectoid point show the same relationship; tensile strength increases. If the whole interval is considered, i.e., \(t_{{{\text{ETo}}}}\), \(\Delta t_{{{\text{liq}}}} - t_{{{\text{ETo}}}}\), and \(\dot{T}\)liq−ETo, the correlation can be explained by several factors, both for the eutectic and eutectoid transformations. For both phase transformations, the cooling rate and the solidification time, respectively, affect the microstructure formation. During the eutectic transformation, a higher cooling rate leads to a finer and more uniform distribution of the graphite spheres in the microstructure. Moreover, in addition to the finer graphite spheres, the austenite grains that form during solidification are also finer. The finer and better distributed graphite spheres lead, among other things, to an increase in tensile strength. The finer austenite structure largely determines the final ferrite or pearlite structure after the solid phase transformation, which is completed after the eutectoid temperature has been exceeded. In both cases, the finer structure is more brittle but stronger, which also explains the observed results.

In the case of eutectoid transformation, a higher cooling rate leads to a higher pearlite content as well as a finer pearlite microstructure resulting from the fine austenite. The finer pearlite microstructure has more grain boundaries, which act as a barrier to dislocation movement and thus increase the strength of the material.18 This also resulted in an increase in tensile strength. Assuming a slower cooling rate, this results in larger graphite spheres during eutectic transformation. And in the eutectoid transformation, a ferrite-containing matrix is formed. This would then also explain the lower hardness and thus the lower tensile strength.

Evaluation of the Observed Correlations and Developed Models

As the investigations show, correlations with the resulting mechanical properties were found for some characteristic temperatures and the corresponding solidification times. The most suitable characteristic values for the explanation were the eutectic temperatures \(T_{{\text{eut,min}}}\), \(T_{{{\text{eut}}}}\) and \(T_{{\text{eut,max}}}\) with coefficients of determination above 0.5 for the tensile strength. The eutectoid temperature, the time intervals from the eutectoid temperature and the end of eutectoid solidification to the eutectoid temperature, and the cooling rate and solidification time from the liquidus temperature to the eutectoid temperature have also been shown to be explanatory variables. In order to link the identified relationships with the mechanical parameters, the thermal modulus was used.

However, mechanical properties do not depend only on a single factor such as a temperature or cooling rate. The main influences of the mechanical properties of ductile cast iron are the chemical composition of the alloy, the cooling rate and the status of nucleation/inoculation of the melt. Although a eutectic temperature can provide information on the chemical composition or cooling rate, it does not solely determine the mechanical properties. It can also be seen from the results that the chemical composition and the resulting eutectic temperature have a higher influence on the mechanical properties of the casting than the cooling time alone.

Thus, if the mechanical parameters are to be calculated with the aid of a model, it makes sense to consider more than one variable, which has been done by multiple regression analysis. With the success that coefficients of determination of up to 0.77 have been achieved. This is a satisfactory result statistically, but raises the question of why it is not even higher, even though the models include multiple descriptive variables. There may be several reasons for this. First, the assumption was made that the characteristic temperatures are identical for all geometries, which is not the case in reality. Second, the results of the descriptive variables were connected via the module. The modulus itself showed little correlation with the mechanical properties. Although it influences the cooling rate, the effect that, for example, a lower cooling rate has on the microstructure and the graphite spheres can be compensated by the chemical composition to a certain degree of the modulus.18,19 Furthermore, the number of data plays an important role. In general, it is likely that significantly more data are needed for a higher coefficient of determination of the models, because the relationship to the mechanical properties of ductile cast iron is very complex. Despite all this, the results of the mechanical properties predicted for the cast are satisfactory, especially for the tensile strength. The average deviation of the tensile strength is 5.84 %.

Conclusions

The aim of this work was to develop an analysis concept that can predict the material properties of spheroidal graphite cast iron by making extended use of thermal analysis data. The material properties investigated were those of tensile strength in particular. In addition, the potential of unused data from thermal analysis was to be demonstrated. This was successfully achieved. The semi-automatic analysis tool developed made it possible to obtain and store the characteristic points of the cooling curve from thermal analyses. The characteristic points included, on the one hand, the characteristic temperatures of the phase transitions and the associated temporal components, such as the cooling rate between the characteristic points.

Supervised learning proved to be a suitable method for analyzing the data. Simple linear regression models were trained with the data obtained from the thermal analysis and the corresponding data of the mechanical characteristic values of geometries with three different wall thicknesses. This allowed conclusions to be drawn about the predictive ability of the characteristic points studied on the respective mechanical parameters of the different geometries. It was found that the temperatures \(T_{{\text{eut,min}}}\), \(T_{{{\text{eut}}}}\) and \(T_{{\text{eut,max}}}\) of the thermal analysis are the most suitable for predicting the tensile strength. Moreover, \(T_{ETo}\) has also shown a correlation for the mentioned parameter. Explanatory relationships have also been found for the temporal component. Thus, the time intervals \(\Delta t_{{\text{eut,end}}} - t_{{{\text{ETo}}}}\) and \(\Delta t_{{{\text{eut}}}} - t_{{{\text{ETo}}}}\), as well as the cooling rates \(\dot{T}\)liq−ETo, \(\dot{T}\)eut,end−ETo and \(\dot{T}\)eut−ETo, have shown an explanatory relationship for the tensile strength of ductile cast iron.

The correlations found were connected with the aid of multiple regression analysis via the thermal modulus of the geometries investigated. A model was derived with a coefficient of determination for the prediction of the tensile strength of 0.77. Using this developed model, a reliable prediction of the mechanical properties could be made with minimal deviation for the respective geometries.

Future work could start at this point. On the one hand, other mechanical properties, such as yield strength, elongation at fracture and uniform elongation or modulus of elasticity, can be considered in the same way. On the other hand, the results of the thermal analysis can be supplemented by further results from the chemical analysis, for example from a spark spectrometer, in order to consider the influence of the alloying elements.

Another problem that could be addressed in building work is the automation of data collection in thermal analysis and the standardization and central storage of all data studied for different casts. This requires a complex information architecture, but would greatly increase the possibility of increasing the knowledge of thermal analysis and its potential to predict various material properties. In addition, with a rapidly growing amount of data, the use of artificial intelligence could become attractive to identify complex data patterns in the large amount of data collected related to thermal analysis.

In summary, the semi-automatic analysis concept, as well as the subsequent supervised learning oriented data analysis, has produced a predictive model for tensile strength. The potential of the thermal analysis cooling curve data analysis has thus been demonstrated, but can be exploited to a much greater extent by increasing the size of the database, through automation and standardization of data collection and storage.