Steel castings are widely used in the manufacturing industry, accounting for approximately 15% of the total casting output in China. They are essential components in machinery, and their quality directly determines the service life and safety of mechanical parts. The quality of steel castings is governed by their microstructure and mechanical properties, which are controlled by casting and heat treatment processes. In this thesis, we present a comprehensive numerical simulation framework that integrates the solidification process and the subsequent heat treatment process to predict the microstructure and mechanical properties of steel castings. The proposed method enables the entire process of steel castings from casting to final heat treatment to be simulated more realistically, thereby providing a basis for optimizing production parameters and improving the quality of steel castings.
Traditional methods for assessing the microstructure and properties of steel castings rely heavily on experimental trial-and-error approaches, which are inefficient, resource-intensive, and difficult to guarantee consistent quality. Numerical simulation has emerged as a powerful alternative, allowing researchers and engineers to visualize the temperature field, microstructure evolution, and property distribution during both casting and heat treatment. However, previous simulation studies usually treated the casting process and the heat treatment process independently. When only the casting process was considered, the as-cast microstructure could not represent the final quality of the component, since subsequent heat treatment is almost always required. When only the heat treatment process was analyzed, the influence of the initial as-cast microstructure was often ignored. To address these shortcomings, this work employs a mapping technique to link the casting solidification process with the heat treatment process, considering the as-cast structure as the initial condition for heat treatment. In this way, the simulation becomes much closer to actual production practice.
The thesis is organized around two major prediction modules. The first module predicts the as-cast grain size, phase fractions, and mechanical properties (yield strength, tensile strength, and hardness) of steel castings after solidification. The second module predicts the phase transformation kinetics and final hardness of steel castings during heat treatment, using the as-cast structure as the initial state. The methods were validated through casting experiments, heat treatment experiments, metallographic analysis, and hardness testing. The validated methods were then applied to a frame-shaped steel casting to demonstrate their usefulness in optimizing the microstructure and performance distribution. Throughout this article, the term steel castings is used extensively to emphasize the focus of this research.
1. Numerical Simulation of the Solidification Process
The accurate prediction of the microstructure and properties of steel castings depends on the precise calculation of the temperature field during solidification. The temperature evolution within the casting and the mold is governed by transient heat conduction, which can be described by the three-dimensional Fourier heat conduction equation:
$$ \rho C_{p} \frac{\partial T}{\partial t} = \lambda \left( \frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} + \frac{\partial^{2} T}{\partial z^{2}} \right) + Q $$
where \(T\) is the temperature, \(t\) is the time, \(\rho\) is the density, \(C_{p}\) is the specific heat capacity, \(\lambda\) is the thermal conductivity, and \(Q\) represents the internal heat source due to the release of latent heat during solidification. The initial conditions are given by assuming that the molten metal fills the mold instantaneously and that the initial temperature distribution is uniform:
$$ T_{\text{cast},0} = T_{1} $$
$$ T_{\text{mold},0} = T_{2} $$
where \(T_{1}\) is the pouring temperature and \(T_{2}\) is the initial mold temperature, usually taken as room temperature. For the boundary conditions, three classic types are considered. In the present work, the third kind of boundary condition, which specifies the interfacial heat transfer coefficient and the ambient temperature, is applied at the casting-mold interface.
The temperature field model is discretized using the finite difference method (FDM). For a three-dimensional problem, a hexahedral unit cell with edge length \(\Delta x\) is considered, and the heat balance is established between the central cell and its six neighboring cells. The energy balance equation can be written as:
$$ \sum_{j=1}^{6} \frac{\Delta t}{\Delta x} \cdot \frac{T_{j}^{t} – T_{i}^{t}}{\frac{\Delta x}{\lambda_{i}} + \frac{\Delta x}{\lambda_{j}}} = \rho_{i} C_{p,i} (\Delta x)^{3} (T_{i}^{t+\Delta t} – T_{i}^{t}) $$
After rearrangement, the temperature of cell \(i\) at the next time step is obtained as a linear combination of the temperatures of the cell and its neighbors at the current time step:
$$ T_{i}^{t+\Delta t} = \left( 1 – \frac{a_{i} \Delta t}{\rho_{i} C_{p,i} \Delta x^{2}} \right) T_{i}^{t} + \sum_{j=1}^{6} \frac{\lambda_{ij} \Delta t}{\rho_{i} C_{p,i} \Delta x^{2}} T_{j}^{t} $$
To ensure numerical stability, the time step must satisfy the following convergence condition:
$$ \Delta t \leq \frac{\rho_{i} C_{p,i} (\Delta x)^{2}}{a_{i}} $$
where \(a_{i}\) is the sum of the thermal conductivities between the central cell and its neighbors. The latent heat released during solidification is treated using several methods, including the enthalpy method, the equivalent specific heat method, and the temperature recovery method. In this work, the equivalent specific heat method is adopted:
$$ C_{p,\text{eq}} = C_{p} – L \frac{\partial f_{s}}{\partial T} $$
where \(L\) is the latent heat and \(f_{s}\) is the solid fraction. By introducing the equivalent specific heat, the heat conduction equation remains in the same form, and the nonlinearity caused by latent heat is handled effectively.
2. Prediction of As-Cast Microstructure and Properties
The prediction of the as-cast microstructure of steel castings includes two major aspects: the grain size distribution and the phase fractions. Grain size is closely related to the nucleation behavior during solidification. In this thesis, we employ the Gaussian continuous nucleation model, which assumes that nucleation occurs on a distribution of heterogeneous nucleation sites. The nucleation site distribution is described by a Gaussian function:
$$ \frac{dn}{d(\Delta T)} = \frac{n_{\max}}{\sqrt{2\pi} \Delta T_{\sigma}} \exp \left[ -\frac{(\Delta T – \Delta T_{N})^{2}}{2(\Delta T_{\sigma})^{2}} \right] $$
where \(n_{\max}\) is the maximum heterogeneous nucleation site density, \(\Delta T_{N}\) is the average nucleation undercooling, and \(\Delta T_{\sigma}\) is the standard deviation of the undercooling. The total number of nuclei at a given undercooling is obtained by integrating the distribution:
$$ n(\Delta T) = \int_{0}^{\Delta T} \frac{dn}{d(\Delta T’)} d(\Delta T’) $$
During each time step \(\Delta t\), the increase in undercooling produces additional nuclei, and the incremental nucleation density is:
$$ \delta n = n(\Delta T + \delta \Delta T) – n(\Delta T) $$
Assuming that each grain grows as a sphere, the grain size \(r\) can be derived from the total nuclei density \(n\):
$$ r = \sqrt[3]{\frac{3}{4\pi n}} $$
This formulation allows the prediction of the grain size of steel castings at every grid point, based on the local cooling history obtained from the temperature field simulation.
For the prediction of phase fractions and mechanical properties, we take advantage of the thermodynamic calculation software JMatPro. This software can compute the phase fractions and mechanical properties of a specific steel composition under various cooling rates, as functions of temperature. By running calculations for a range of cooling rates from 0.03°C/s to 15°C/s, we establish a comprehensive database for the phase fractions and mechanical properties of various steel castings. The database includes more than 100 steel grades, such as ZG15, ZG20CrMo, ZG25, ZG30Mn, ZG40Mn, ZG45, ZG55, ZG65Mn, and ZG270-500. An example of the calculated data for ZG20Mn at a cooling rate of 5°C/s is illustrated in the following table:
| Temperature (°C) | Austenite (%) | Ferrite (%) | Pearlite (%) | Bainite (%) | Martensite (%) | Hardness (HV) |
|---|---|---|---|---|---|---|
| 1450 | 100.0 | 0.0 | 0.0 | 0.0 | 0.0 | — |
| 800 | 18.5 | 61.2 | 20.3 | 0.0 | 0.0 | 142 |
| 600 | 0.0 | 36.5 | 63.5 | 0.0 | 0.0 | 175 |
| 400 | 0.0 | 36.5 | 63.5 | 0.0 | 0.0 | 178 |
To predict the phase fractions and mechanical properties of an actual casting, the cooling rate for each grid cell must be determined from the temperature field simulation. The solidus and liquidus temperatures are used to calculate the time at which the grid enters and exits the mushy zone. The average cooling rate in the solid-liquid region is then obtained as:
$$ V = \frac{T_{\text{liquidus}} – T_{\text{solidus}}}{t_{2} – t_{1}} $$
where \(t_{1}\) and \(t_{2}\) are the times corresponding to the liquidus and solidus temperatures, respectively. Once the cooling rate is known, the database is searched for the corresponding phase fractions and mechanical properties at the target temperature. A fast data search algorithm and a grid assignment technique were developed to efficiently map the database results onto the finite difference mesh, resulting in the spatial distribution of phase fractions and mechanical properties across the entire steel casting.
This approach effectively overcomes the difficulty of determining uncertain parameters that are often required in more fundamental models. By relying on JMatPro calculations, the database provides consistent thermodynamic and kinetic data, while the temperature field from the finite difference simulation provides the necessary thermal history information. The combination of these tools allows for a robust prediction of the as-cast state of steel castings.
3. Prediction of Heat Treatment Microstructure and Properties
Heat treatment is a critical step in controlling the final microstructure and mechanical properties of steel castings. In this work, we propose a model that predicts the phase transformation kinetics and hardness of steel castings during heat treatment. A key feature of the model is that the as-cast microstructure obtained from the solidification simulation is mapped onto the heat treatment grid as the initial state. This mapping is achieved by retaining the casting cells and replacing the mold cells with the heating or cooling medium, as illustrated conceptually in this work.
During heat treatment, the temperature field is again calculated using the finite difference method. The boundary condition at the surface of the steel casting involves both convection and radiation heat transfer. The combined boundary condition is expressed as:
$$ -\lambda \frac{\partial T}{\partial n} = H_{k} (T_{w} – T_{c}) + \sigma \varepsilon (T_{w}^{4} – T_{c}^{4}) $$
where \(H_{k}\) is the convective heat transfer coefficient, \(T_{w}\) is the surface temperature of the workpiece, \(T_{c}\) is the temperature of the heating or cooling medium, \(\sigma\) is the Stefan-Boltzmann constant, and \(\varepsilon\) is the surface emissivity. For simplicity, this is often linearized as:
$$ -\lambda \frac{\partial T}{\partial n} = H (T_{w} – T_{c}) $$
where \(H\) is the total heat transfer coefficient.
The phase transformation kinetics during heat treatment are calculated using the TTA (time-temperature-austenitization) curve for the heating stage and the TTT (time-temperature-transformation) curve for the cooling stage. Since the actual heat treatment process is non-isothermal, the continuous heating or cooling curve is discretized into a series of small isothermal steps, and the superposition rule is applied. The incubation period for each phase transformation is determined by accumulating the ratio of the time step to the isothermal incubation period:
$$ \sum_{i=1}^{n} \frac{\Delta t_{i}}{\tau_{i}(T_{i})} = 1 $$
When the sum reaches unity, the transformation begins. The transformation fraction is then calculated using the Avrami equation for diffusional transformations:
$$ f = 1 – \exp(-b t^{n}) $$
where \(b\) and \(n\) are temperature-dependent coefficients. These coefficients are determined by fitting the isothermal transformation kinetics at selected temperatures using the TTT curve data. For two known time points \(t_{1}\) and \(t_{2}\) corresponding to transformed fractions \(V_{1}\) and \(V_{2}\), the exponent \(n\) is calculated as:
$$ n = \frac{\ln \left[ \frac{\ln(1-V_{1})}{\ln(1-V_{2})} \right]}{\ln \left( \frac{t_{1}}{t_{2}} \right)} $$
and the coefficient \(b\) is obtained from:
$$ b = -\frac{\ln(1-V_{1})}{t_{1}^{n}} $$
For non-isothermal conditions, the transformation fraction at the current time step is computed using the fictitious time method. If the transformation fraction at the previous step is \(f_{i-1}\), the fictitious time \(t^{*}\) required to achieve this fraction at the current temperature is:
$$ t^{*} = \left[ -\frac{\ln(1-f_{i-1})}{b_{i}} \right]^{1/n_{i}} $$
where \(b_{i}\) and \(n_{i}\) are the coefficients at the current temperature \(T_{i}\). The new transformation fraction is then:
$$ f_{i} = 1 – \exp \left[ -b_{i} (t^{*} + \Delta t_{i})^{n_{i}} \right] $$
This procedure is repeated for each time step until the transformation is complete.
For martensitic transformation, which is diffusionless and depends only on temperature, the Koistinen-Marburger equation is used:
$$ f_{M} = 1 – \exp[-\alpha (M_{s} – T)] $$
where \(\alpha\) is a constant, typically taken as 0.011 for carbon steels, \(M_{s}\) is the martensite start temperature, and \(T\) is the current temperature. The incremental martensite fraction at each time step is:
$$ \Delta f_{M,i} = f_{A} \left\{ 1 – \exp[-\alpha (M_{s} – T_{i})] \right\} – f_{M,i-1} $$
where \(f_{A}\) is the retained austenite fraction at the start of martensite transformation.
After the phase fractions are determined, the final hardness of the steel casting is calculated using the single-phase hardness weighted average method:
$$ Hv = V_{M} Hv_{M} + V_{B} Hv_{B} + V_{F} Hv_{F+P} + V_{P} Hv_{F+P} $$
where \(V_{M}\), \(V_{B}\), \(V_{F}\), and \(V_{P}\) are the volume fractions of martensite, bainite, ferrite, and pearlite, respectively. The hardness of each single phase is expressed as a function of alloy composition and the cooling rate at 700°C. The relevant equations are:
$$ Hv_{B} = -323 + 185C + 330Si + 153Mn + 65Ni + 144Cr + 191Mo + \log_{10}(V_{r}) (89 + 53C + 55Si – 22Mn – 10Ni – 20Cr – 33Mo) $$
$$ Hv_{F+P} = 42 + 223C + 53Si + 30Mn + 12.6Ni + 7Cr + 19Mo + \log_{10}(V_{r}) (10 – 19Si + 4Ni + 8Cr + 130V) $$
$$ Hv_{M} = 127 + 949C + 27Si + 11Mn + 8Ni + 16Cr + 21\log_{10}(V_{r}) $$
where \(V_{r}\) is the cooling rate at 700°C in degrees Celsius per hour. By computing the cooling rate from the temperature field simulation, the hardness of each grid cell can be evaluated, leading to a spatial hardness distribution map for steel castings after heat treatment.
4. Experimental Validation
To verify the proposed prediction methods, experiments were carried out on a steel casting. The material used was ZG270-500, a widely used medium-carbon cast steel with good combination of strength and toughness. The chemical composition is listed in the following table:
| C (%) | Si (%) | Mn (%) | S (%) | P (%) | Ni (%) | Cr (%) | Cu (%) | Mo (%) | V (%) |
|---|---|---|---|---|---|---|---|---|---|
| 0.32 | 0.40 | 0.51 | 0.035 | 0.035 | 0.23 | 0.18 | 0.30 | 0.14 | 0.05 |
The thermophysical properties of the steel and the mold materials are summarized in the following tables:
| Property | Value |
|---|---|
| Density (g/cm³) | 7.8 |
| Thermal conductivity (cal/cm·s·°C) | 0.0548 |
| Specific heat (cal/g·°C) | 0.1543 |
| Latent heat (cal/g) | 55 |
| Liquidus temperature (°C) | 1500 |
| Solidus temperature (°C) | 1439 |
| Material | Density (g/cm³) | Thermal conductivity (cal/cm·s·°C) | Specific heat (cal/g·°C) |
|---|---|---|---|
| Mold (furan resin sand) | 1.56 | 0.00192 | 0.258 |
| Chill | 7.86 | 0.115 | 0.124 |
| Insulating sleeve | 0.80 | 0.0008 | 0.15 |
| Air | 0.00121 | 0.000062 | 0.24 |
The casting process was simulated using the software developed in our laboratory. The temperature distribution at various times was obtained, and the cooling curves at several measurement points were extracted. The simulated cooling curves were then compared with the actual temperature measurements obtained from thermocouples embedded in the mold during the pouring experiment. The results showed good agreement between the simulated and measured temperature histories, confirming the accuracy of the temperature field model.
After the casting was cooled to room temperature, metallographic specimens were taken from the riser area. The specimens were prepared using standard grinding and polishing techniques and then examined under an optical microscope. The as-cast microstructure of the ZG270-500 steel casting consisted primarily of ferrite, pearlite, and bainite, depending on the local cooling rate. The grain size was measured using image analysis, and the ferrite fraction was estimated from the metallographic images. The measured values were compared with the simulation predictions, as shown in the following table:
| Specimen | Measured grain size (μm) | Simulated grain size (μm) | Measured ferrite (%) | Simulated ferrite (%) |
|---|---|---|---|---|
| 1 | 155.72 | 165.78 | 40.28 | 36.56 |
| 2 | 93.31 | 96.54 | 24.09 | 27.81 |
The comparison shows that the simulated grain sizes and ferrite fractions are close to the measured values, with minor deviations attributed to image analysis errors and simplifications in the numerical model. The hardness values were also measured using a Vickers hardness tester. The comparison between measured and simulated hardness is presented below:
| Specimen | Measured hardness (HV) | Simulated hardness (HV) |
|---|---|---|
| 1 | 165 | 175 |
| 2 | 187 | 205 |
These results confirm that the proposed solidification prediction method can reliably estimate the as-cast grain size, phase fractions, and hardness of steel castings.
For the heat treatment validation, small test bars were machined from the riser area of the same steel casting. The test bars were heated to 870°C in a muffle furnace, held for sufficient time to achieve complete austenitization, and then cooled in air or water. After heat treatment, the specimens were examined metallographically and their hardness was measured. The air-cooled specimen exhibited a microstructure of ferrite, pearlite, and bainite, while the water-quenched specimen exhibited a fully martensitic structure. The simulated phase fractions and hardness values are compared with the experimental results in the following table:
| Specimen | Cooling medium | Simulated phases | Measured hardness (HV) | Simulated hardness (HV) |
|---|---|---|---|---|
| 3 | Air | Ferrite, pearlite, bainite | 261 | 247 |
| 4 | Water | Martensite | 521 | 534 |
The simulated hardness values are within 5.4% of the measured values, demonstrating that the heat treatment prediction model accurately captures the phase transformation behavior and the resulting hardness of steel castings.
5. Application to Frame-Shaped Steel Castings
After validation, the proposed methods were applied to a frame-shaped steel casting with significant variations in wall thickness. The casting geometry included thick sections, thin bars, and risers, which created large differences in cooling rates and therefore large variations in microstructure and mechanical properties.
The temperature field simulation revealed that the thin bars cooled much faster than the thick sections. Consequently, the as-cast microstructure varied significantly across the casting. The thin bars exhibited a high fraction of martensite and bainite, resulting in high hardness and strength, while the thick sections exhibited a mixture of ferrite, pearlite, and bainite with lower hardness and strength. The simulated microstructure and mechanical properties at five characteristic points are summarized in the following table:
| Point | Martensite (%) | Bainite (%) | Ferrite (%) | Pearlite (%) | Hardness (HRC) | Yield strength (MPa) | Tensile strength (MPa) |
|---|---|---|---|---|---|---|---|
| 1 | 3.17 | 81.41 | 15.29 | 0.13 | 23.46 | 591 | 833 |
| 2 | 0.00 | 69.79 | 27.09 | 3.12 | 14.52 | 472 | 693 |
| 3 | 0.00 | 75.32 | 23.48 | 1.20 | 16.50 | 494 | 721 |
| 4 | 14.60 | 71.62 | 13.67 | 0.05 | 31.30 | 740 | 995 |
| 5 | 49.91 | 41.50 | 8.46 | 0.01 | 46.53 | 1211 | 1462 |
It is evident that the mechanical properties of the frame-shaped steel casting are highly non-uniform, which would severely limit its service performance. To improve the uniformity and overall quality, two heat treatment processes were evaluated.
In the first process, the casting was heated to 870°C, held to achieve complete austenitization, and then water-quenched. The simulation predicted that the final microstructure would consist predominantly of martensite and bainite, with high hardness values above 420 HRC and good wear resistance. However, the toughness would be relatively low. In the second process, the casting was austenitized and then air-cooled. The simulation predicted a microstructure of ferrite, pearlite, and bainite, with hardness values ranging from 180 to 200 HRC, resulting in better toughness but lower wear resistance.
The application demonstrates that the proposed prediction framework can effectively evaluate different heat treatment processes and assist engineers in selecting the optimal treatment for steel castings based on the specific service conditions. For components requiring high wear resistance, water quenching is recommended, while air cooling is suitable for components subjected to impact loading. The ability to simulate both the solidification and heat treatment processes in a unified framework provides valuable insights for optimizing the manufacturing route of steel castings.
6. Conclusion and Outlook
In this thesis, a comprehensive numerical simulation framework was developed for the prediction of microstructure and mechanical properties of steel castings throughout the solidification and heat treatment processes. The main conclusions are summarized as follows:
First, a temperature field model was established and discretized using the finite difference method. The model incorporates boundary conditions and latent heat effects, and the numerical stability condition was derived. The accurate temperature field forms the basis for all subsequent microstructure and property predictions.
Second, a solidification prediction method was proposed. The Gaussian continuous nucleation model was used to calculate the nucleation density and grain size of steel castings. A comprehensive database of phase fractions and mechanical properties was established using JMatPro software. A fast data search algorithm and grid mapping technique were developed to obtain the spatial distribution of phase fractions, yield strength, tensile strength, and hardness in steel castings.
Third, a heat treatment prediction model was developed. The as-cast microstructure from the solidification simulation was mapped onto the heat treatment domain as the initial state. The superposition rule was applied to discretize continuous heating and cooling into small isothermal steps. The TTA and TTT curves were used to calculate incubation periods and transformation fractions. The hardness was predicted using the single-phase hardness weighted average method.
Fourth, the proposed methods were validated through casting experiments, heat treatment experiments, metallographic analysis, and hardness testing. The simulated temperature curves, grain size, phase fractions, and hardness values were in good agreement with the experimental measurements, confirming the accuracy and reliability of the proposed methods for steel castings.
Finally, the prediction framework was applied to a frame-shaped steel casting. The simulation successfully captured the large variations in microstructure and mechanical properties caused by the complex geometry. The evaluation of two heat treatment processes showed that the framework can guide the selection of appropriate heat treatment parameters to achieve the desired performance distribution in steel castings.
Several aspects of the present work can be improved in the future. The cooling rate calculation during solidification should be further optimized by increasing the frequency of temperature field output. The database of phase fractions and mechanical properties should be expanded to cover a wider range of steel grades and cooling conditions. Furthermore, the heat treatment simulation should incorporate stress-strain coupling to account for distortion and residual stress, which are important for large and complex steel castings. The development of a fully coupled thermal-metallurgical-mechanical model would significantly enhance the accuracy and applicability of the prediction framework.
In conclusion, the numerical simulation framework developed in this thesis provides a powerful tool for predicting the microstructure and mechanical properties of steel castings throughout the entire manufacturing process. The integration of solidification and heat treatment simulation enables a more realistic and holistic approach to process optimization. The validated methods offer significant practical value for the casting industry, helping to improve the quality and performance of steel castings while reducing development time and cost.