The control of dimensional accuracy in investment casting has long been a critical challenge that restricts the improvement of product quality. The lengthy process chain, encompassing wax pattern preparation, shell sintering, and alloy solidification, introduces cumulative deviations with significant genetic effects, making it difficult to precisely control the final casting dimensions. Our research addresses this issue by constructing a comprehensive numerical simulation model that covers the entire process of investment casting. This study systematically investigates the deformation behavior and error transmission mechanisms through the wax pattern mold filling, shell sintering, and alloy solidification stages, providing a theoretical basis for dimensional accuracy control in investment casting.
1. Introduction
Investment casting, also known as lost-wax casting, offers unique advantages in manufacturing complex structural parts and has become a key near-net-shape forming process for high-end equipment core components in the aerospace and other industries. However, the long process flow with numerous procedures, such as wax pattern injection, shell building and sintering, and alloy pouring and solidification, introduces cumulative dimensional deviations. These deviations exhibit a significant genetic effect, rendering the precise control of final casting dimensions a persistent problem. Numerical simulation technology provides an important means for understanding the deformation mechanism and optimizing process parameters through the quantitative description and prediction of multi-physics field behavior during the casting process. Current research in this field has evolved from traditional single-process analysis to multi-field coupling and has progressed from single-stage simulation to full-process integrated simulation covering wax pattern forming, shell sintering, and alloy solidification. This advancement provides strong support for improving casting quality.
Wax pattern preparation, as the initial step in investment casting, has its dimensional accuracy directly determining the size of the assembled wax cluster. Studies have shown that dimensional deviations introduced during the wax pattern stage can account for more than 40% of the final casting dimensional deviations, highlighting the importance of controlling the source of the dimensional transmission chain. The wax pattern mold filling process involves non-Newtonian fluid flow and transient heat transfer. Models like the Cross-WLF model can effectively describe the pseudoplastic rheological behavior of the wax material. Subsequent shell sintering is a crucial step for ensuring cavity dimensional accuracy, and numerical simulation has become an important tool for revealing its deformation mechanism. The final solidification stage, which determines the casting’s dimensional accuracy, widely employs multi-physics field coupled simulations. Our study systematically reviews the current state of numerical simulation for these three core stages: wax pattern mold filling, shell sintering, and alloy solidification.
While current research demonstrates that numerical simulation methods for individual processes have become mature and can accurately simulate the process, most studies focus on the local optimization of independent processes, lacking a holistic perspective. The data transmission between process stages is missing, preventing the formation of a complete full-process simulation system and making it difficult to systematically trace the evolution of casting deformation behavior. Therefore, conducting in-depth research on full-process numerical simulation, building a “wax pattern-shell-casting” full-process simulation platform to break the data barriers along the process chain, revealing the genetic mechanism of dimensional accuracy along the process chain, and establishing corresponding reverse deformation compensation models are of significant theoretical value and engineering application importance for improving casting forming quality and dimensional accuracy in investment casting.
2. Full-Process Numerical Simulation Models for Investment Casting
2.1 Wax Pattern Injection Simulation Model
We simulated the wax filling and packing process using a dedicated simulation software. The built-in Cross-WLF viscosity model was selected to characterize the viscosity behavior of the wax material. To obtain accurate model parameters, parameter fitting for the Cross-WLF model was performed based on experimental data from wax rheological property measurements.
The Cross-WLF viscosity model is mathematically expressed as follows:
Where:
| Symbol | Description |
|---|---|
| $\eta$ | Melt viscosity |
| $\eta_0$ | Zero-shear viscosity |
| $\dot{\gamma}$ | Shear rate |
| $\tau^*$ | Critical stress at the transition to shear-thinning behavior |
| $n$ | Power-law index at high shear rates |
The zero-shear viscosity $\eta_0$ is a function of temperature and pressure:
Where the glass transition temperature $T^*$ and the parameter $A_2$ are pressure-dependent:
$A_1$, $\tilde{A_2}$, $D_1$, $D_2$, $D_3$ are material fitting coefficients, $T$ is temperature, and $p$ is pressure.
The wax material is a thermorheological material whose modulus exhibits time-temperature dependence. The total stress is decomposed into deviatoric and spherical stress tensors. The stress-strain relationship is described using an integral-type constitutive equation for investment casting simulations:
Where:
| Symbol | Description |
|---|---|
| $\sigma_{ij}$ | Total stress tensor |
| $\delta_{ij}$ | Kronecker delta |
| $e_{ij}$ | Deviatoric strain tensor |
| $\epsilon_m$ | Volumetric strain |
| $\epsilon_{th}$ | Thermal strain ($\epsilon_{th} = \alpha \Delta T$, $\alpha$ is thermal expansion coefficient) |
| $G(\xi – \xi’)$ | Shear relaxation modulus |
| $K(\xi – \xi’)$ | Bulk relaxation modulus |
| $\xi, \xi’$ | Reduced time variables |
2.2 Shell Sintering Simulation Model
We developed a custom numerical simulation program to simulate the shell sintering process in investment casting. This process is treated as a transient heat conduction problem, including a heating phase due to an external heat source. The temperature field in the Cartesian coordinate system is governed by:
Where:
| Symbol | Description |
|---|---|
| $\rho$ | Density ($kg/m^3$) |
| $C_p$ | Specific heat capacity ($J/(kg \cdot °C)$) |
| $\lambda$ | Thermal conductivity ($W/(mm \cdot °C)$) |
| $L$ | Latent heat |
| $f_s$ | Solid fraction |
We employed the widely used Skorohod-Olevsky Viscous Sintering (SOVS) model, a continuum mechanics model based on porous viscoplastic rheological theory. This model captures sintering behavior phenomenologically, reducing the number of required model parameters. The model defines shear viscosity $\eta_s$ and bulk viscosity $\eta_b$:
Where $\theta$ is the porosity (ratio of pore volume to total volume), and $\eta$ is the apparent viscosity of the fully dense solid. The temperature dependence of $\eta$ is described by the Arrhenius viscosity model for investment casting shell materials:
Where $Q_V$ is the surface activation energy, $R$ is the gas constant, $T$ is the absolute temperature, and $\eta_0$ is a material constant. The main driving force for sintering, $\sigma_s$, originates from the surface tension of the material’s pores:
Where $\gamma_s$ is the surface tension and $G$ is the average grain size.
2.3 Casting Solidification Simulation Model
We used a custom numerical simulation program for the investment casting solidification process. For the thermo-elastic-plastic model, the total strain increment consists of elastic, plastic, and thermal strain increments:
The elastic strain increment is:
Where $G$ is the shear modulus, $d\sigma’_{ij}$ is the deviatoric stress increment, $\nu$ is Poisson’s ratio, $E$ is the elastic modulus, and $d\sigma_m$ is the hydrostatic pressure increment.
The plastic strain increment, based on the Prandtl-Reuss flow rule, is:
Where $d\bar{\epsilon}^p$ is the equivalent plastic strain increment, $\bar{\sigma}$ is the equivalent stress, and $\sigma’_{ij}$ is the deviatoric stress.
The thermal strain increment is:
The incremental stress-strain relationship is:
3. Full-Process Numerical Simulation Analysis for Investment Casting
3.1 Wax Pattern Injection Simulation
We analyzed the source and transmission of dimensional errors throughout the investment casting process by tracking the three stages: wax pattern preparation, shell sintering, and casting solidification. A three-way pipe component was selected as the study object. The simulation for wax pattern injection was performed using a non-Newtonian fluid model. The key parameters for the simulation are summarized below:
| Parameter | Value |
|---|---|
| Geometry | Three-way pipe, longest section 166 mm |
| Wax Material | K512 |
| Injection Temperature | 55 °C |
| Injection Time | 6 s |
| Injection Pressure | 22 MPa |
The flow pattern comparison between simulation and actual wax injection experiments showed good consistency, validating the reliability of the numerical model for investment casting. The simulation accurately reproduced the flow evolution at the cavity boundaries and effectively characterized the filling offset caused by the gating system design. To verify the dimensional accuracy of the wax pattern, the actual wax pattern was scanned using a coordinate measuring machine (CMM). The simulated deformation results were compared with the scanned model using 3D surface registration. The absolute deformation at each point was calculated as:
Where $u$, $v$, and $w$ are the deformation components in the X, Y, and Z directions, respectively.
The verification results for the wax pattern stage are shown in the following table:
| Metric | Value |
|---|---|
| Average surface simulation error | 0.518 mm |
| Verification method | CMM 3D surface registration |
| Model reliability | Validated for wax filling process representation |
3.2 Shell Sintering Numerical Simulation
Based on the wax pattern injection simulation results, we conducted numerical simulation of the shell sintering process in investment casting. A geometric model of the shell was constructed by extracting the outer surface of the assembled wax pattern model to generate a shell solid model representing the actual process dimensions. A stepped heating curve was used for the sintering process to ensure complete binder removal and suppress thermal stress cracking.
The simulation captured the temperature field evolution during sintering. Key observations include:
- A distinct temperature difference was observed between the outer surface (point A) and inner surface (point B) of the shell, which is a major cause of deformation.
- During the heating phase, points expanded in the negative X direction, followed by a reversal around 12,000 seconds due to binder phase transformation, leading to an overall shrinkage trend.
- Maximum shrinkage was reached at the beginning of the holding stage. Subsequent displacement was primarily related to high-temperature creep and cooling shrinkage.
To validate the shell cavity deformation, the actual shell was scanned using industrial Computed Tomography (CT) with a spatial resolution of 111 µm. The internal cavity geometry was reconstructed from the CT point cloud data. The simulated cavity deformation was extracted using numerical interpolation methods based on the initial wax pattern model and shell sintering results. The comparison results are as follows:
| Metric | Value |
|---|---|
| Average surface simulation error | 0.705 mm |
| Verification method | Industrial CT scanning & 3D surface registration |
| Model reliability | Validated for shell sintering process representation |
3.3 Casting Solidification Numerical Simulation
For the casting solidification stage, the geometric model was built by extracting the deformed internal cavity of the shell from the sintering simulation results. The pouring temperature was set to 1700 °C. The simulation was performed on the full cluster, which included the gating system and multiple cavities. The displacement field of the target casting was isolated for analysis. To verify the simulation, the actual casting was scanned using a CMM to reconstruct its 3D digital model.
The surface error between the simulated casting deformation and the scanned actual casting was analyzed using 3D surface registration. The results are summarized below:
| Metric | Value |
|---|---|
| Average surface simulation error | 0.760 mm |
| Verification method | CMM scanning & 3D surface registration |
| Model reliability | Validated for casting solidification deformation prediction |
3.4 Full-Process Deformation Inheritance and Error Genetic Mechanism
We performed a cross-process data calibration using the initial wax pattern mold cavity as the geometric benchmark. The deformations from the wax pattern stage were used as input for the shell sintering simulation, and the resulting shell cavity deformation was used as input for the casting solidification simulation. This allowed us to trace the cumulative deformation through the “wax pattern-shell-casting” process chain of investment casting.
To quantitatively characterize the deformation inheritance, a specific circular cross-section on the casting was selected for analysis. The design value for this section was 103.0 mm, and the design diameter of the wax pattern cavity was 105.5 mm. The simulated and measured diameters at each stage are presented in the following table:
| Process Stage | Simulated Diameter (mm) | Measured Diameter (mm) |
|---|---|---|
| Wax Pattern Mold Filling Stage | 104.72 | 104.71 |
| Shell Sintering Stage | 102.72 | 103.33 |
| Alloy Solidification Stage | 103.19 | 104.01 |
The analysis shows excellent agreement between simulation and measurement, with relative errors for local feature sizes in each key process stage controlled within 1%. This validates the predictive accuracy of the numerical simulation model. The evolution of the section diameter reveals a non-monotonic error genetic mechanism in investment casting:
- Shrinkage occurs during the wax pattern mold filling stage.
- A more significant shrinkage of the shell cavity occurs during the shell sintering stage due to the sintering of ceramic materials.
- During the alloy solidification stage, the casting diameter rebounds from the shell cavity diameter. This reflects the combined effect of the metal liquid cooling phase change and solidification shrinkage against the already contracted shell constraint, demonstrating a dynamic compensation effect between sequential processes in investment casting.
To further elucidate the error genetic mechanism, we extracted surface errors relative to both the initial mold cavity and the final product. The confidence intervals for the errors are summarized as follows:
| Comparison Basis | Process Stage | Cumulative Deviation (mm) |
|---|---|---|
| Simulation vs. Product | Wax Pattern Stage | 0.518 |
| Shell Sintering Stage | 0.705 | |
| Alloy Solidification Stage | 0.760 | |
| Simulation vs. Initial Mold Cavity | Wax Pattern Stage | 0.532 |
| Shell Sintering Stage | 4.213 | |
| Alloy Solidification Stage | 3.766 |
Compared to the initial mold, the cumulative deformation shows a non-monotonic transfer characteristic in investment casting. The error increases sharply from 0.532 mm to 4.213 mm during shell sintering due to ceramic material shrinkage. Subsequently, during alloy solidification, the solidification shrinkage occurring within the already contracted shell cavity causes the final casting deviation to decrease to 3.766 mm. This confirms the genetic enhancement of errors along the process flow, revealing the deformation inheritance through the “wax pattern mold filling-shell sintering-alloy solidification” multi-stage processes. The shell sintering stage is identified as the most critical link, accounting for over 87% of the total full-process deformation. This finding suggests that the shell sintering stage should be the primary target for dimensional accuracy control in investment casting. It also highlights the potential for utilizing the dynamic compensation mechanism between processes for optimized process design.
4. Conclusion
Based on full-process numerical simulation and experimental validation for investment casting, this study systematically investigated the deformation behavior and error transmission mechanisms across the wax pattern mold filling, shell sintering, and alloy solidification stages. The main conclusions are as follows:
(1) A full-process deformation numerical simulation method covering wax pattern mold filling, shell sintering, and alloy solidification in investment casting was established. The reliability of the single-process models and the full-process deformation inheritance simulation was validated, with average surface simulation errors for the wax pattern, shell, and casting being 0.518 mm, 0.705 mm, and 0.760 mm, respectively, all below 0.8 mm.
(2) A dynamic compensation mechanism and error genetic characteristics between the multiple stages of investment casting were revealed. The study demonstrates that cumulative deviation significantly increases during the shell sintering stage due to high-temperature shrinkage of ceramic materials. However, the subsequent solidification shrinkage of the alloy within the contracted cavity provides a reverse compensation effect, causing the final casting deviation to decrease. This confirms the genetic enhancement of errors along the process flow.
(3) A multi-scale validation method based on numerical interpolation and 3D surface registration was proposed. By integrating CMM and industrial CT scanning data to reconstruct physical models, this method enables the correlation analysis of macroscopic displacement fields and geometric accuracy throughout the investment casting process. This enhances the credibility of the simulation results while providing an effective means for mold cavity design compensation and process parameter optimization, offering significant engineering value for achieving precise dimensional control of castings.