**Authors:** M.I. Terentiev, O.A. Broytman, E.E. Chikinova, A.V. Monastyrski

**Keywords:** Heat treatment modeling, steel, quenching, tempering, mechanical properties, microstructure.

This article presents a method for analyzing heat treatment processes of steel parts, with the prediction of mechanical properties and microstructure, using models implemented in the "PoligonSoft" computational simulation system. Using the example of the automated implementation of quenching and tempering technology development in preparation for the production of a new type of product, the methodology for adjusting the modeling tool to the conditions of the available thermal equipment is demonstrated, and the general process of the computational study is described.

The integration of a thermal solver into computational simulation systems for foundry processes (CSS FP) is a fundamental requirement to enable the numerical modeling of casting formation processes. This feature of CSS FP generally allows for solving various heat transfer tasks, including those beyond the specificity of metallurgical production.

In particular, the advanced and adjustable thermal solver of the "PoligonSoft" CSS FP enables the analysis of temperature variations in bodies that are in static or dynamic contact, interacting with the environment and adjacent bodies through radiative and convective heat transfer, while considering transformations occurring in the material, among other factors. These adjustment capabilities of the thermal solver, including its integration with a solver for stress-strain state analysis, allow for an appropriate coupled numerical analysis of the conditions of heat treatment (HT).

The observation of temperatures, stresses, and deformations in the part already provides valuable information for the heat treatment technologist, but it is insufficient for a clear interpretation and conclusion about the outcome of the heat treatment (HT).

As a natural development of the "PoligonSoft" CSS FP in this context, a module for calculating the structure and properties of metal parts after HT has been incorporated, allowing the numerical analysis cycle to be closed with an automatic forecast of the final result. This includes the visualization of possible defects such as cracks and dimensional deviations exceeding allowed tolerances, and provides the distribution of microstructural characteristics and mechanical properties.

The integration of the HT simulation tool into the technological development process, as is often the case, requires certain adaptation measures. It is advisable to leverage the accumulated experience in the production of heat-treated parts, recorded in the form of measured characteristics of structure and properties in parts of various configurations and masses, to adjust the CSS and ensure the correct formulation of the problem and the input of appropriate initial data in the analysis of HT conditions for the planned new products. The methodology for such adjustment must be efficient, allowing full consideration of the specific characteristics of the technological process implementation on the available production equipment.

The HT simulation using "PoligonSoft" is carried out using a 3D finite element (FE) model of the steel part. Key data on temperature distribution variations throughout the volume of the part are obtained by numerically solving the transient heat conduction equation, with boundary conditions that depend on the stage of the process (heating, cooling, contact with other parts, tooling, local influence, etc.).

An accurate prediction of the structure and properties after HT can be achieved with statistical models based on a representative sample. The HT module includes such a model, based on the "Creusot-Loire" calculation methodology (named after the developer company) [1, 2]. This methodology includes:

A model of austenite decomposition, through which the microstructure and hardness are calculated once the polymorphic transformation is complete during steel cooling.

A tempering model, which calculates the hardness field upon completion of the operation.

The methodology is suitable for analyzing HT processes that include heating to austenitizing temperature and subsequent cooling (conditions for annealing, normalizing, quenching), as well as for heating and holding for high-temperature tempering. In addition to calculating hardness and microstructural characteristics during HT simulation, statistical models are available for calculating ultimate tensile strength (σ_{в}), conventional yield strength (σ_{0,2}), and relative elongation (δ_{5}).

Calculations using this model can be applied to low-carbon, medium-carbon, and low-alloy steels with alloying element contents as detailed in Table 1.

Table 1

**Figure 1** presents a general scheme of the model's applicability in relation to the chemical compositions of steels from all grades included in the following Russian national standards (GOST) for metal products:

GOST 1050-2013 "Metal products from unalloyed structural quality and special steels. General technical requirements";

GOST 19281-2014 "High-strength rolled products. General technical requirements";

GOST 21357-87 "Castings made of cold-resistant and wear-resistant steel. General technical requirements";

GOST 4543-2016 "Metal products made of alloyed structural steels. Technical requirements";

GOST 977-88 "Steel castings. General technical requirements."

**Figure 1. **Applicability of the model to steel grades according to different standards:

The diagram was constructed using the following principle (see legend): "no" – the content of at least one chemical element is outside the allowed range; "partially" – at least one element has content that only partially falls within the allowed range; "completely" – the ranges of all chemical elements are within the allowed values.

The procedure for conducting the computational analysis and the method for visualizing the result can be demonstrated through a numerical experiment simulating a hardenability test (Figure 2a) according to GOST 5657-69 "Steel. Hardenability test method" using the end-quench (Jominy) method for 38ХГМ steel according to GOST 4543-2016.

**Figure 2. **Hardenability test of steel: **a** – process scheme, **b **– computational model configuration with input of initial data.

Water cooling

Sample

Fixation point in the device

In the calculations (Figure 2b), a finite element geometric model of a standard cylindrical sample with a diameter of 25 mm and a height of 100 mm is used. The coupled analysis of thermal processes and structural transformation, along with the formation of the set of mechanical properties, requires the input of non-geometric data, such as the thermophysical properties of the steel, the actual chemical composition of the steel in the casting, and the specific conditions of the heating process for quenching: temperature, isothermal holding time, and boundary conditions in different areas (forced water cooling at the end and air cooling on the remaining surfaces).

As a result of the calculations, it is possible to observe the variation in the temperature field over time in the sample from the initial 860°C, as well as the cooling rate field throughout its volume (Figures 3a and 3b), and the distribution of the steel's structure and properties. Given the problem setup, the primary focus of the study is the prediction of hardness beneath the surface layer along the height of the sample (Figure 3d).

**Figure 3. **Main results of the computational experiment:**a **and

The microstructure of an alloy with a given chemical composition explains the origin of its properties, so when analyzing the results, it is advisable to consider these types of forecasts together. For instance, in the area adjacent to the end, which has undergone cooling at extreme rates, there is a relatively low variation in hardness due to the high proportion of martensite, or even with 100% martensite content (Figure 3e).

**Figure 4. **Hardenability curve - **1** and hardness along the height of the sample after tempering - **2**, based on the results of calculations (solid lines) and physical tests (dots, dashed line).

In **Figure 4**, the hardness calculation result is presented using the integrated SSC tool for graph creation, in the form of a hardenability curve compared with available experimental data for the studied steel composition. The hardness distribution along the height, obtained through tempering simulation of a quenched sample, is also shown. This type of analysis is not part of standard testing but can be used by technologists to evaluate the influence of a tempering regime on the final properties of the piece, based on different initial hardness values.

The maximum relative error in predicting hardness values for a quenched sample is less than 10%, and for a quenched and tempered sample, around 6%.

During the performance of heat treatment (TT), defects such as cracks, warping, or dimensional inaccuracies may form. The analysis of the stress-strain state during the simulation cycle allows, at the technology development stage, an assessment of the effectiveness of measures taken to reduce the tendency for the appearance of the mentioned defects. **Figure 5** shows some results of these calculations for oil quenching of a 40X steel part.

**Figure 5. **Stress-strain state calculation during the quenching of a shaft: **a** – dimensional requirements;

To achieve greater accuracy in predicting the structure and properties of metal parts after HT when integrating the corresponding analysis tool into the technological development process, a series of adaptation measures must be carried out. Some of these measures are quite straightforward: training the personnel of the technology department in the use of the SSC, modifying the IT infrastructure to account for the incorporation of a new digital tool, enhancing the integrated database on the properties of alloys and other materials used in production, updating company standards regarding instructions that regulate the development process using numerical calculations, among other actions.

A key part of the adaptation process is the verification of the calculation results, using the existing experience in the production of heat-treated parts, with the implementation of measures to minimize discrepancies between predictions and practical outcomes. As a result of this type of "fine-tuning," it will be possible, in the future, to take into account less obvious aspects of equipment operation and process implementation in production, incorporating the necessary corrections into the parameters of the numerical models to be introduced as initial data.

Let us consider a situation that corresponds to the described conditions. Data had been accumulated in the production process regarding the HT regimes and mechanical properties of quenched and tempered parts made from different steel grades. The requirement was to develop, using the SSC, and then implement within the existing capabilities, the HT for a new part made from a new steel grade, ensuring that the selected quenching and tempering regimes would achieve the required mechanical properties.

Table 2

The data on the geometric parameters of the pieces, the tempering and quenching regimes, and the mechanical properties are presented in **Table 2**, while the actual chemical composition of the metal in the heats is detailed in **Table 3**.

Tabla 3

Tensile tests were carried out in accordance with GOST 1497-84, with samples for testing taken from 1/3 of the radius from the surface in the case of cylindrical parts and from half the wall thickness for plate-type parts and hollow cylinders. Among the listed elements, the ring made of 36Х2Н2МФА steel represented a new type of product for which it was necessary to develop an appropriate HT regime through simulation.

The ring stands out for its dimensions compared to previously processed parts; generally, it would be preferable to study the manufacturing conditions of products of the same type. However, this is not always possible. The case we are considering is suitable for a comparative analysis due to the relatively similar volumetric and mass characteristics between the new part and those already subjected to HT in practice. The composition and quality of the quenching medium are the same, and quenching will be carried out in the same quenching tank. Therefore, it can be assumed that the cooling conditions for the new parts will be similar to those of the previous ones, which implies the identity of the heat transfer coefficient α at the metal-quenching medium interface. Given the extreme cooling rates during quenching, the result of the numerical calculation is highly sensitive to the heat transfer coefficient introduced, making the use of an appropriate value a key factor in ensuring satisfactory accuracy in the simulation results.

The determination of an adjusted heat transfer coefficient for oil quenching through a series of computational experiments is, in this case, the action that will largely allow the adaptation of the computational model of the process to existing production conditions. In this and other HT calculations, the model integrated into "PoligonSoft," based on the "Creusot-Loire" methodology, was used.

A simple trial and error method was applied: according to the known HT regimes, quenching and tempering simulations were performed for parts whose resulting properties were already known (parts 2...4). For each part, a certain number of calculation cases were considered, in each of which different heat transfer coefficients at the metal-quenching medium interface were assigned. The predicted temporary tensile strength was recorded in the areas from which samples for physical tests were extracted, allowing a comparison of the calculated and actual values. The root mean square error (RMSE) of the tensile strength calculation was chosen as the optimization metric.

Where σ_{вi} and σ^{р}_{вi} represent, respectively, the actual and calculated values of the tensile strength in the i-th test; *n* is the number of available tensile strength values (here *n = 6*).

To reduce the number of calculations and types of analysis, it is reasonable to make use of existing knowledge on the behavior of steels during heat treatment (HT). In particular, the HT analysis of the steel shaft part made of 18ХГТ included only the quenching simulation, which is entirely acceptable, since low-temperature tempering has an insignificant effect on the strength properties of steels whose martensitic transformation start temperature is higher than that of tempering. For 18ХГТ steel, this temperature is approximately 350°C [3], so it can be concluded that tempering already occurs partially during cooling [4].

Figure 6 shows the graph of the RMSE variation as a function of α. It was determined that the value that minimizes the calculation error is α = 1000 W/(m² K), which fits well within the known ranges for oil quenching conditions [5]. This value should be used in HT calculations for the new product. In this case, the sample size is relatively small, primarily serving to demonstrate the principle, which explains the relatively high value of the minimum RMSE. Whenever possible, it is advisable to use larger samples.

**Fig. 6. **Root mean square error (RMSE) of the tensile strength calculation (σв) as a function of the heat transfer coefficient

The complete cycle of numerical analysis of the HT conditions for the production of the new product type includes:

Calculation of the heating process in the furnace to determine the necessary holding time;

Simulation of quenching and tempering, with variations in the HT regimes to identify the range of technological process parameters that ensure the required properties.

In addition to the use of verified heat transfer coefficients in thermal calculations (established, for example, using the methodology described earlier), it is important to emphasize the critical need to introduce temperature-dependent thermophysical properties of the material. These can be obtained from the built-in database in the SSC, from bibliographic sources, calculated using specialized software tools, or determined through in-house experimental work when feasible.

According to the technological instructions used by the company, the calculation of the heating duration for steel parts must follow commonly accepted empirical dependencies [6]. During the simulation of the complete HT cycle, the selection of the holding time during heating for quenching and tempering was primarily based on established practices, while simultaneously investigating the possibility of optimization.

The results of the simulation of the ring heating process for quenching in an electric pit furnace are shown in Figure 7 (as part of the HT cycle). According to the typical technological process and recommendations for alloyed steels, the load for quenching heating is carried out at a temperature of 650...750°C, followed by additional furnace heating to the target austenitizing temperature of 860°C, after which isothermal holding is performed. The furnace load consists of a single part.

**Fig. 7. **Change in the temperature of the piece (ring) over time during the execution of different operations in the heat treatment cycle: **1** and **2** – curves for the control points at the center and surface, respectively; **3** – absolute temperature difference between the surface and the center; **4** and **5** – furnace temperature during heating for quenching and tempering, respectively.

The calculation of the critical point for the completion of the austenitic transformation according to different models [7, 8] results in A_{c3} = 800°C. The results of the simulation for quenching heating indicate that A_{c3} is reached quickly after the furnace regime stabilizes (approximately 1 hour after the part is introduced into the furnace), and approximately 1.5 additional hours are required to heat up to the austenitizing temperature, ensuring uniform temperature distribution across the section. The total holding time is 4 hours.

The temperature changes during the quenching process are also shown in Figure 7. The thermal calculations were combined with the analysis of the structure and properties during quenching and tempering. The main objective of the computational study was to determine the appropriate tempering temperature by varying it across a series of calculations. Figure 8 presents a nomograph summarizing the predictions of the mechanical characteristics σ_{в} and σ_{0.2} (in the section of the ring from which samples were taken for testing) for different tempering temperatures. It is clear from the figure that to ensure the required set of properties, tempering must be carried out at a temperature of 600...660°C. Considering the need for a safety margin for the tensile strength, as well as the expected error of the model for strength properties of approximately 50 MPa (see Figure 6), it is recommended to test tempering within the narrower temperature range of 620...630°C.

**Fig. 8. **Results of the calculations of tensile strength and yield strength (solid lines) in the piece (ring) made of 36Х2Н2МФА steel as a function of tempering temperature, compared with the mechanical property requirements (dashed lines).

The tempering regime established in this way is presented in the resulting temperature-time cycle from the calculation, as shown in Figure 7. The uniformity of temperatures across the section of the part and the achievement of the tempering temperature occur approximately 4 hours after the part is loaded into the furnace. The total time spent in the furnace, from the moment of loading, is considered to be 6.5 hours.

According to the results of the simulation of the temperature-time cycles of the HT, it is reasonable to assume the need for additional experimental work at the plant to refine and optimize the heating regimes.

Table 4

The quenching and tempering regime developed through the simulation of HT processes was implemented in practice. Table 4 presents its main parameters, as well as the mechanical properties measured in the physical tests of the samples taken from the controlled area of the ring material. The result of the calculation of the distribution of mechanical properties throughout the entire volume of the piece (i.e., in samples that can be taken from different parts of the steel ring) is illustrated in Figure 9. A good agreement between the calculation results and the experimental data is shown.

**Fig. 9. **Prediction of the mechanical property characteristics in the piece (ring, section) after quenching and tempering: **a** – yield strength (left) and tensile strength (right), MPa; **b** – relative elongation, %.

The error observed in the predicted values indicates a satisfactory accuracy of the simulation and the success of the measures taken to adapt the process to the existing thermal production conditions, particularly the adequacy of using the constant value of the established heat transfer coefficient.

The analysis of the results of the microstructure prediction for the heat-treated ring helps explain the formation of different levels of mechanical properties at various stages of the process (Figure 10). In general, it is recommended to combine the analysis of the results of the microstructure, properties, and the cooling rate field in the volume of the piece, which can be correlated with the results of the physical hardenability tests of the steel, using the tools integrated into the SSC. As a result of quenching in the case analyzed, a bainitic-martensitic structure is observed, with a bainite proportion of 50-90% in most of the ring section, and a high proportion of martensite, on average ~70%, only at the surface and the edges of the piece, where the cooling rates are highest.

**Fig. 10. **Prediction of the microstructure distribution (%) after quenching of the ring (section):

**a** – bainitic, **b** – martensitic, **c** – ferritic-pearlitic.

The new digital analysis tool for HT processes, which complements the functionality of the SSC "PoligonSoft," enables the automation of the development of quenching, normalization, annealing, and tempering technologies for steel parts. The process of adapting the tool to the existing plant conditions, in an environment with appropriate work discipline, is quite transparent and involves the use of available records on HT regimes, as well as the monitoring of mechanical properties and the structure of previously produced parts. The amount of data obtained through simulation regarding the development of HT processes and their outcomes significantly exceeds what is typically obtained through physical testing, which facilitates and accelerates the search for optimal technological solutions in production, while also being of great value in research work.

[1] Maynier P., Jungmann B., Dollet J. Creusot-Loire System for Predicting the Mechanical Properties of Low-Alloy Steel Products, Hardenability Concepts with Applications to Steel // The Metallurgical Society of AIME, 1978, pp. 518–545.

[2] Maynier P., Dollet J., Bastein P. Influence of Alloying Elements on the Hardenability of Low-Alloy Steels // Revue de Metallurgie, vol. 67, №4, April, 1970, pp. 343–351.

[3] Popov A.A., Popova L.E. Isothermal and Thermokinetic Diagrams of Supercooled Austenite Decomposition. Heat Treater's Handbook. Moscow: Metallurgy, 1965. 495 p.

[4] Fédin V.M., Ronzhina Yu.V., Borz A.I., Vakulenko S.P., Tikhonov D.P., Ushakov B.K. Thermal Strengthening of Elastic Elements of Railway Rolling Stock by High-Speed Water Flow. Moscow: VINITI RAS, 2012. 236 p.

[5] Smoljan B., Iljkic D., Totten G.E. Mathematical Modeling and Simulation of Hardness of Quenched and Tempered Steel. Metallurgical and Materials Transactions B, vol. 46, 2015, pp. 2666–2673.

[6] Firger I.V. Heat Treatment of Alloys. Leningrad: Mashinostroenie, 1982. 304 p.

[7] Kasatkin O.G., Vinokur B.B., Timoshenko V.L. Calculation Models for Determining Critical Points of Steel // MiTOM, 1984, №1, pp. 20–22.[8] Trzaska J., Dobrzanski L.A. Modeling of CCT Diagrams for Engineering and Construction Steels // Journal of Materials Processing Technology, 2007, №192–193, pp. 504–510.

Translated by A.J. Camejo Severinov

Original text in Russian

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