Precision casting, widely recognized as investment casting, is a near-net-shape manufacturing technology that enables the mass production of components with high dimensional accuracy, excellent surface quality, and complex geometry. It improves material utilization, reduces subsequent machining operations, and significantly saves cost and time. It is extensively applied in aerospace, automotive, medical devices, and electronic power industries. In recent years, precision casting technology has been increasingly used for the manufacture of large, complex, thin-walled structural components, which places higher demands on the process level of this technology.
Wax pattern preparation is the starting process of precision casting. The dimensional error of the wax pattern accumulates and transfers to the next generation as the process advances, ultimately affecting the dimensional accuracy of the casting. The wax pattern preparation process is complex and long, and the forming process includes filling and packing, in-mold cooling, demolding, and post-demolding deformation stages. In the present work, I comprehensively considered the viscoelastic deformation characteristics of each stage of wax pattern forming and conducted a numerical simulation study on the stress and deformation of the entire process of “filling and packing – in-mold cooling – demolding – post-demolding deformation”.
1. Introduction and Research Background
The precision casting process mainly includes pattern assembly, casting, and post-processing, which can be subdivided into wax injection, wax tree assembly, slurry dipping and sand stuccoing, shell dewaxing, shell roasting, alloy pouring, shell shaking, and cutting. Each process step produces dimensional errors that are inherited by the next generation. The dimensional errors generated in the precision casting process mainly include: manufacturing error of the wax pattern die, shrinkage deformation during wax injection, warpage deformation during wax pattern storage, dimensional error caused by thermal expansion during shell dewaxing and roasting, shrinkage deformation during alloy pouring and solidification, and dimensional deformation during heat treatment and hot isostatic pressing.
Because dimensional errors accumulate and transfer with the process, monitoring and controlling the dimensional deformation of each step is a difficult problem, which makes the die design of the wax pattern particularly difficult. To meet the dimensional accuracy requirements of castings, it is necessary to deeply understand the characteristics of wax, shell, and metal materials, study the dimensional deformation mechanism of each process flow, and the law of dimensional error accumulation and transfer in the process.
The preparation of the wax pattern is the starting point of the precision casting process and directly determines the dimensional shape of the shell. The dimensional instability of the wax pattern causes casting dimensional errors. In precision casting, the shrinkage deformation of the wax pattern has a great influence on the dimensional accuracy of the final casting, and the quality of the wax pattern significantly affects the dimensional accuracy of the casting. To ensure the production and manufacture of precision castings, the dimensional deformation of the wax injection molding process must be controlled.
The main flow of wax injection molding is to align the wax injection nozzle with the die gate, then close the die and inject the wax. After the wax fills the cavity, the packing stage begins. After the wax pattern cools in the die for a period, the die is opened and the pattern is removed. Wax injection molding may produce defects such as pores, sink marks, cracks, surface wrinkles, and deformation. The linear shrinkage of the wax pattern is small, and the overall deformation is limited. Cracks and sink marks can be identified by observation alone, while dimensional deformation can only be found through actual measurement. Due to the characteristics of viscoelastic materials, the wax pattern continues to deform after demolding.
In actual production, operators often judge the forming quality of the wax pattern only by the dimensional measurement results after demolding and cooling, lacking real-time monitoring and precise control of the dimensional deformation during the wax pattern storage stage. As a result, wax patterns whose dimensional accuracy does not meet the requirements enter the subsequent precision casting processes, affecting the quality of the final casting.
Traditional wax injection molding usually requires multiple trial molds and mold repairs to ensure the dimensional accuracy of the pattern. This “trial-and-error” method consumes a large amount of manpower, material resources, and financial resources, and the experimental cycle is long and the cost is high. Moreover, due to the limitations of experimental conditions, it is often impossible to monitor the dimensional deformation of the entire process flow of wax pattern forming in real time. With the gradual maturity of computer technology, numerical simulation technology has become an important means to study wax injection molding. It can assist in the process design of wax injection molding, shorten the research and development cycle, reduce the cost of repeated trial and error, achieve precise control of wax pattern forming dimensions, and ensure the performance and quality of precision castings.
To address the above problems, I used numerical simulation to simulate and analyze the stress and deformation of the entire wax injection molding process, explored the dimensional deformation law of the whole wax pattern preparation process, studied the influence law of process parameters on wax pattern forming quality, and carried out process parameter optimization analysis, providing scientific guidance for the design and optimization of wax injection molding processes in precision casting.
2. Theoretical Models for Wax Injection Molding
Wax injection molding mainly includes filling, packing, in-mold cooling, and post-demolding storage stages. In the filling stage, the wax rapidly fills the cavity under injection pressure. Thereafter, because the melt temperature decreases, packing is required to compensate for the volume shrinkage of the wax caused by changes in pressure and temperature. When the gate solidifies, continuing to apply packing pressure has no effect on feeding, and the already shaped wax pattern continues to cool in the die until it is removed. The demolded wax pattern is usually not directly used for shell making; instead, it is left to stand for a period after cooling to room temperature, and the subsequent precision casting processes are carried out after the wax pattern dimensions stabilize.
2.1 Viscosity Constitutive Model
The wax filling process involves the non-Newtonian flow of a viscous fluid and the non-isothermal, unsteady heat transfer between the wax and the die. To ensure smooth filling of the wax, the viscosity of the wax selected for precision casting is generally relatively high, so as to avoid turbulence and splashing and to ensure that the wax fills the mold in a laminar flow manner. During filling, the temperature, pressure, and shear rate of the wax vary over a wide range. Unlike Newtonian fluids, the viscosity of the wax is not only related to temperature but also to shear rate and pressure. The filling process of the wax pattern is an unsteady flow of the wax over a wide temperature range and shear rate range, and the viscosity decreases with increasing shear rate.
I selected the Cross-WLF model as the viscosity model of the wax to simulate the filling and packing process. The Cross-WLF model considers the effect of pressure on viscosity and can accurately reflect the rheological behavior of the wax over a wide range of temperatures and shear rates. The relationship is as follows:
$$\eta = \frac{\eta_0}{1 + \left(\frac{\eta_0 \dot{\gamma}}{\tau^*}\right)^{1-n}}$$
where $\eta$ is the melt viscosity, $\eta_0$ is the zero-shear viscosity, $\dot{\gamma}$ is the shear rate, $\tau^*$ represents the critical stress level for the transition to shear thinning, and $n$ represents the power-law index in the high shear rate method. The zero-shear viscosity $\eta_0$ can be expressed as:
$$\eta_0 = D_1 \exp\left[-\frac{A_1(T-T^*)}{A_2 + (T-T^*)}\right]$$
where $T$ is the temperature, $T^*$ is the glass transition temperature determined by curve fitting, $A_2 = \tilde{A}_2 + D_3 p$ with $p$ being the pressure, and $D_1$, $A_1$, $\tilde{A}_2$, and $D_3$ are data fitting coefficients. The glass transition temperature is expressed as $T^* = D_2 + D_3 p$, where $D_2$ is a data fitting coefficient.
2.2 Viscoelastic Constitutive Model
An ideal elastic solid obeys Hooke’s law when subjected to an external force, and the stress and strain remain in phase at all times. The strain is independent of time, and after the external force is unloaded, the strain recovers instantaneously and the elastic deformation is reversible; its deformation has no time dependence. When a certain force is applied to a viscoelastic material, its deformation changes with time, and the recovery process after unloading the force has a certain delay. Its characteristic is that the strain lags behind the stress, and the response of the strain to the stress is not completed instantaneously; that is, the mechanical behavior of a viscoelastic material has a certain time dependence.
Wax and other polymers have viscoelastic characteristics. Compared with elastic materials, the thermal residual stress of the wax pattern during cooling relaxes partly, and the warpage after demolding is also reduced. Thereafter, the wax pattern continues to produce viscoelastic deformation during storage until the dimensions stabilize. To accurately describe the viscoelastic behavior of the wax, an appropriate viscoelastic constitutive model must be selected. Assuming that the stress response after a constant strain is applied is expressed as:
$$\sigma(t) = Y(t)\,\sigma_0$$
where $Y(t)$ is the relaxation modulus. The above equation can represent the stress relaxation behavior of a product after a constant strain is applied. However, in actual injection molding, the load borne by the product is not a constant strain but is complex and time-varying. To describe this problem, the Boltzmann superposition principle is required, which considers that the strain is a function of the entire history, and the contributions of each stress to the strain are independent, and the total strain is a linear superposition of these stresses.
Let the strain generated by the object under external action be $\varepsilon(t)$, which varies with time. Using the relaxation modulus $Y(t)$, the stress formula can be obtained according to the superposition principle:
$$\sigma(t) = \int_{-\infty}^{t} Y(t-\xi)\frac{d\varepsilon(\xi)}{d\xi}\,d\xi$$
I used the generalized Maxwell model to characterize the viscoelastic properties of the wax pattern. It is composed of one spring and several Maxwell fluid models connected in parallel. For a model with multiple Maxwell units connected in parallel, the tensile relaxation modulus is:
$$E(t) = E_e + \lim_{n\to\infty}\sum_{i=1}^{n} E_i e^{-t/\tau_i} = E_e + \int_{0}^{\infty} F(\tau)e^{-t/\tau}\,d\tau$$
where $E_e$ is the tensile elastic modulus in the steady state. The above equation is often expressed in logarithmic coordinates as:
$$E(t) = E_e + \int_{0}^{\infty} H(\tau)e^{-t/\tau}\,d\ln\tau$$
where $H(\tau)$ is called the relaxation time spectrum, and $H(\tau)d\ln\tau$ represents the contribution to stiffness between $\ln\tau$ and $\ln\tau + d\ln\tau$. In the shear case, it can be expressed as:
$$G(t) = G_{\infty} + \int_{0}^{\infty} H(\tau)e^{-t/\tau}\,d\ln\tau$$
where $G_{\infty}$ is the shear relaxation in the steady state, and $H(\tau)$ is the relaxation time spectrum. Similarly, the creep compliance can be obtained as:
$$J(t) = J_{\infty} + \int_{-\infty}^{\infty} L(\tau)e^{-t/\tau}\,d\ln\tau$$
where $J_{\infty}$ is the elastic compliance in the steady state and $L(\tau)$ is the retardation time spectrum. When studying the viscoelastic properties of materials, the relaxation time spectrum and the retardation time spectrum are two important functions.
2.3 Time-Temperature Superposition Principle
With temperature fluctuations, the viscoelasticity of polymers also changes. Materials that are difficult to creep under normal temperature conditions exhibit more obvious stress relaxation and creep phenomena in high-temperature environments. From a microscopic point of view, the polymer molecular chains are stretched by external forces, forming internal resistance. Under the thermal motion effect of the polymer chains, some entanglement points are untied, the polymer chains slip, and the curled state is reshaped, so the internal stress decreases. The recombination of polymer chains and the slip of molecular chains are the main reasons for stress relaxation and creep of polymers. Temperature has a significant effect on the molecular thermal motion of polymers. At high temperatures, the internal friction of the polymer chains is small, and stress relaxation is fast. After the temperature drops, the stress relaxation effect is also slowed down due to the increase in internal friction resistance.
The viscoelasticity of polymers is not only related to temperature but also shows time dependence. The relaxation behavior that needs to be observed over a longer time at low temperature can be observed in a shorter time at high temperature. This is the time-temperature superposition principle, that is, the viscoelasticity of polymers can be mutually converted in time and space dimensions. According to this principle, the modulus curve of the wax pattern at a certain reference temperature can be obtained by shifting and superposing the modulus curves at other temperatures. The time-temperature superposition principle can be expressed as:
$$E(t,T) = E(t/\alpha_T, T_0)$$
where $T_0$ is the reference temperature and $\alpha_T$ is the shift factor in the time-temperature superposition principle. For amorphous polymers, the shift factor $\alpha_T$ can be described by the WLF equation:
$$\log \alpha_T = -\frac{C_1(T-T_0)}{C_2 + T – T_0}$$
where $C_1$ and $C_2$ are fitting parameters. When the temperature exceeds the applicable range of the WLF equation, the Arrhenius model can be used:
$$\ln \alpha_T = \frac{E_a}{R}\left(\frac{1}{T} – \frac{1}{T_0}\right)$$
where $E_a$ is the apparent activation energy obtained by experimental fitting, and $T_0$ is the reference temperature.
2.4 Three-Dimensional Viscoelastic Deformation Model
For a three-dimensional viscoelastic material, considering the stress contribution $d\sigma_{ij}(t)$ of the strain increment $d\varepsilon_{ij}(\xi)$ at time $t$, with $t > \xi$, we obtain:
$$d\sigma_{ij}(t) = Y_{ijkl}(t-\xi)\,d\varepsilon_{kl}(\xi)$$
Integrating gives:
$$\sigma_{ij}(t) = \int_{-\infty}^{t} Y_{ijkl}(t-\xi)\frac{d\varepsilon_{kl}(\xi)}{d\xi}\,d\xi$$
where $Y_{ijkl}$ is a fourth-order tensor, and $\xi$ is the reduced time, $\xi = \int_{0}^{t} 1/\alpha_T\,d\lambda$. If the material is isotropic, for an elastic body we have $\sigma_{ij} = 2G\varepsilon_{ij} + \lambda\delta_{ij}\varepsilon_{kk}$. The wax belongs to a thermorheologically simple material, and the modulus of the material is a function of time and temperature. An integral constitutive equation is used to describe the stress-strain relationship of the wax pattern:
$$\sigma(t) = \int_{-\infty}^{t} G(\xi-\xi’)\frac{\partial e_{ij}(t)}{\partial t}\,dt + \delta_{ij}\int_{-\infty}^{t} K(\xi-\xi’)\frac{\partial(\varepsilon_m(t)-\varepsilon_{th})}{\partial t}\,dt$$
The material stress can be divided into a deviatoric stress tensor and a spherical stress. The deviatoric stress tensor is expressed as:
$$\sigma_{ij} = \int_{-\infty}^{t} G(\xi-\xi’)\frac{\partial e_{ij}(t)}{\partial t}\,dt$$
The spherical stress tensor is expressed as:
$$\sigma_m = \int_{-\infty}^{t} K(\xi-\xi’)\frac{\partial(\varepsilon_m(t)-\varepsilon_{th})}{\partial t}\,dt$$
where $G(\xi-\xi’)$ is the shear relaxation modulus of the material, $K(\xi-\xi’)$ is the bulk relaxation modulus, and $\varepsilon_{th}$ is the strain caused by thermal expansion or contraction, with $\varepsilon_{th} = \alpha\Delta T$, where $\alpha$ is the linear thermal expansion coefficient. The shear relaxation modulus of the generalized Maxwell model is expressed as:
$$G(t) = G_{\infty} + \sum_{i=1}^{n} G_i \exp(-t/\tau_i)$$
where $G_{\infty}$ is the shear relaxation modulus after complete relaxation of the material, and $G_i$ and $\tau_i$ are the shear relaxation modulus and relaxation time of the Maxwell components, respectively. Similarly, the bulk relaxation modulus is expressed as:
$$K(t) = K_{\infty} + \sum_{i=1}^{n} K_i \exp(-t/\tau_i)$$
In viscoelastic calculations, it is generally considered that the bulk modulus remains constant during relaxation. If the Poisson’s ratio is constant, the change trend of the shear modulus is consistent with that of the elastic modulus.
3. Material Characterization and Parameter Fitting
I selected K512 wax as the injection material. The thermal properties, rheological properties, PVT relationship, Poisson’s ratio, and other parameters of the wax were measured, and the relaxation time spectrum of the wax was determined and fitted. The technical indicators of the wax are summarized in Table 1.
| Parameter | Value |
|---|---|
| Needle penetration (DMM) | 7 |
| Ash content (%) | 0.022 |
| Linear shrinkage (%) | 0.9–1.1 |
3.1 Thermal Properties
The thermal properties of the wax mainly refer to thermal conductivity, specific heat capacity, and transition temperature. The cooling process of the wax pattern is a three-dimensional unsteady heat transfer process, and both the thermal conductivity and specific heat capacity of the wax are related to temperature. To accurately simulate the temperature and heat transfer behavior of the wax pattern forming process, it is necessary to measure the variation curves of these two parameters with temperature. In the wax injection molding simulation, the transition temperature is an important criterion for judging whether the melt has solidified. When the wax temperature is lower than the transition temperature, solidification occurs. The measured transition temperature of the wax was 49 °C.
I measured the thermal conductivity of the wax in both solid and liquid states using a transient plane source instrument. Meanwhile, the specific heat capacity of the K512 wax was measured over a temperature range of 30–70 °C. The wax has very poor thermal conductivity, which may differ from that of alloy liquid by more than two orders of magnitude. The measured thermal conductivity and specific heat curves provide the necessary input for the subsequent simulation.
3.2 Rheological Properties and Viscosity Fitting
I selected the Cross-WLF model to describe the rheological properties of the wax during filling. The dynamic viscosity of the K512 wax was measured at 50, 54, 58, 61, and 64 °C using the steady-state mode of a rheometer. The fitted Cross-WLF model parameters are listed in Table 2.
| Parameter | Value | Unit |
|---|---|---|
| $n$ | 0.3691 | – |
| $\tau^*$ | 0.427887 | Pa |
| $D_1$ | 2.18 × 1036 | Pa·s |
| $D_2$ | 263.15 | K |
| $D_3$ | 0 | K/Pa |
| $A_1$ | 123.92 | – |
| $A_2$ | 51.6 | K |
3.3 Poisson’s Ratio
The Poisson’s ratio refers to the ratio of the transverse normal strain to the axial normal strain when the material is subjected to uniaxial tension or compression, reflecting the characteristics of the transverse deformation of the material. A universal mechanical testing machine was used to complete the measurement of the stress-strain curve and Poisson’s ratio during the tensile process. The test temperature was room temperature, and the loading rate of the instrument was 1 mm/min. The Poisson’s ratio of the wax was taken as 0.47.
3.4 PVT Behavior and Fitting
The PVT relationship describes the relationship between the specific volume of the polymer and temperature and pressure. It is a basic characteristic of polymers and can reflect information such as density, bulk modulus, and thermal expansion coefficient during the material forming process. It is very important for analyzing the physical changes occurring in the injection process and predicting the shrinkage deformation of products, and it is a necessary parameter for injection molding numerical simulation.
The modified dual-domain Tait state equation is widely used to describe the relationship between the specific volume of the polymer melt and the pressure and temperature fields during injection molding. The expression is as follows:
$$V(T,P) = V_0(T)\left\{1 – C\ln\left[1 + \frac{P}{B(T)}\right]\right\} + V_1(T,P)$$
where $V(T,P)$ is the specific volume under temperature $T$ and pressure $P$, $V_0$ is the specific volume at zero pressure, $C$ is a universal constant generally taken as 0.0894, and $B$ is the pressure sensitivity coefficient of the material. According to the thermodynamic properties of polymers, the PVT relationship needs to be described by two temperature domains, as follows. When $T < b_5 + b_6 P$:
$$
\begin{cases}
V_0 = b_{1s} + b_{2s}(T – b_5) \\
B(T) = b_{3s}\exp[-b_{4s}(T – b_5)] \\
V_1 = b_7\exp[b_8(T – b_5) – b_9 P]
\end{cases}
$$
When $T > b_5 + b_6 P$:
$$
\begin{cases}
V_0 = b_{1m} + b_{2m}(T – b_5) \\
B(T) = b_{3m}\exp[-b_{4m}(T – b_5)] \\
V_1 = 0
\end{cases}
$$
where $b_5$ represents the transition temperature of the measured volume at zero pressure, and $b_6$ represents the change trend of the transition temperature with pressure. The transition temperature increases linearly due to pressure, and the remaining coefficients are obtained by curve fitting.
The linear thermal expansion coefficient can be derived from the modified dual-domain Tait state equation. From the equation, the isothermal compression coefficient $\kappa_T$ is:
$$\kappa_T = -\frac{dV/V}{dP} = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T = \frac{C}{(P+B)[1 – C\ln(1+P/B)]}$$
The linear thermal expansion coefficient $\alpha$ is:
$$\alpha = \frac{1}{3}\cdot\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P = \frac{1}{3}\left(\frac{b_2}{V_0} – P\kappa_T b_4\right)$$
I measured the PVT relationship of the wax using an isobaric cooling method, with test pressures of 4, 6, 8, and 10 MPa and a test temperature range of 40–70 °C. The inflection points in the PVT experimental data were extracted first, and $b_5$ and $b_6$ were determined by linear regression. Then, a sequential quadratic programming algorithm was used to perform nonlinear regression on the experimental data of the solid and molten parts separately. Both coefficients of determination were greater than 0.99, indicating the reliability of the fitting results. The fitted coefficients of the modified dual-domain Tait state equation are listed in Table 3.
| Coefficient | Value | Unit |
|---|---|---|
| $b_{1m}$ | 0.00114 | m3/kg |
| $b_{2m}$ | 1.372 × 10-6 | m3/(kg·K) |
| $b_{3m}$ | 6.4005 × 107 | Pa |
| $b_{4m}$ | 0.1685 | 1/K |
| $b_{1s}$ | 0.001141 | m3/kg |
| $b_{2s}$ | 3.478 × 10-6 | m3/(kg·K) |
| $b_{3s}$ | 3.9838 × 107 | Pa |
| $b_{4s}$ | 0.0296 | 1/K |
| $b_5$ | 324.15 | K |
| $b_6$ | 5 × 10-7 | K/Pa |
| $b_7$ | 0 | m3/kg |
| $b_8$ | 0 | 1/K |
| $b_9$ | 0 | 1/Pa |
3.5 Relaxation Time Spectrum
An ideal elastic body is described by Hooke’s law, where stress and strain are linearly related, and an ideal viscous body is described by Newton’s law, where stress and strain rate are linearly related. Hooke’s law and Newton’s law need to satisfy certain conditions to hold, that is, the strain or strain rate is sufficiently small. The properties of linear viscoelasticity are between those of an ideal elastic body and an ideal viscous fluid, and the condition for the establishment of linear viscoelasticity is also deformation under small strain. The linear shrinkage of the K512 wax is small, and the wax injection molding process is constrained by the cavity or core, so the deformation of the wax pattern forming process belongs to small deformation, which can be described by a linear viscoelastic constitutive model.
To accurately describe the viscoelastic behavior of the wax pattern, it is necessary to measure the linear viscoelastic region of the material, carry out mechanical property parameter measurement experiments within this region, and fit the constitutive model parameters. The linear viscoelastic region was obtained by amplitude scanning using a rotational rheometer. The dynamic moduli corresponding to different strains were plotted, and the region where the modulus remained basically unchanged was the linear viscoelastic region of the measured sample. Within a strain rate of 0.1%, the storage modulus and loss modulus were basically unchanged. Although a smaller strain still lies in the linear viscoelastic region, an excessively small strain amplitude easily causes measurement errors of the instrument, so a strain rate of 0.1% was selected as the standard for subsequent dynamic modulus testing.
The wax flow conforms to the characteristics of a thermorheologically simple material. The viscoelastic property curves at different temperatures can be shifted to the same curve in logarithmic coordinates, that is, time and temperature are equivalent, and the shift size of the curve can be measured using the shift factor. The injection temperature of the K512 wax is around 60 °C, and the temperature of the wax during the injection molding process is basically near or below the transition temperature, so the Arrhenius model was selected. The dynamic moduli of the wax were measured at 30, 35, 45, 50, and 55 °C using the rotational oscillation mode. The experiment was carried out under an alternating force field, and increasing the temperature and decreasing the frequency are equivalent. With 45 °C selected as the reference temperature, the test results were shifted to a single curve, and the shift factor was fitted based on the Arrhenius model. The coefficient of determination of the fitting was 0.999.
According to the linear viscoelastic theory, the shear relaxation modulus of the generalized Maxwell model in the frequency domain is expressed as follows:
$$G_s(\omega) = G_{\infty} + \sum_{i=1}^{n}\frac{G_i\tau_i^2\omega^2}{1+\tau_i^2\omega^2} = G_0\left(1-\sum_{i=1}^{n}g_i\right) + G_0\sum_{i=1}^{n}\frac{g_i\tau_i^2\omega^2}{1+\tau_i^2\omega^2}$$
$$G_l(\omega) = \sum_{i=1}^{n}\frac{G_i\tau_i\omega}{1+\tau_i^2\omega^2} = G_0\sum_{i=1}^{n}\frac{g_i\tau_i\omega}{1+\tau_i^2\omega^2}$$
where $G_s(\omega)$ and $G_l(\omega)$ are the storage modulus and loss modulus in the shear oscillation mode, $G_0$ is the instantaneous shear relaxation modulus, $G_{\infty}$ is the shear relaxation modulus in the steady state, $G_i$ and $\tau_i$ represent the shear modulus and relaxation time of a single element in the generalized Maxwell model, respectively, and $g_i$ is the weighting coefficient, $g_i = G_i/G_0$.
When dealing with linear viscoelastic problems, the bulk relaxation modulus is generally considered not to change with time and is only related to temperature. Therefore, solving the relaxation time spectrum means fitting the shear modulus weighting coefficient and relaxation time of each Maxwell element, expressed as $(g_i, 0, \tau_i)$. Based on the sequential quadratic programming algorithm, nonlinear regression was performed on the storage modulus and loss modulus of the K512 wax in the frequency domain. It was found that the fitting was relatively accurate when the number of Maxwell elements was 15, and the coefficient of determination of the fitting was 0.999. The normalized relaxation time spectrum of the K512 wax is listed in Table 4.
| No. | $g_i$ | $k_i$ | $\tau_i$ |
|---|---|---|---|
| 1 | 0.249872 | 0 | 3.05 × 10-6 |
| 2 | 0.108686 | 0 | 4.17 × 10-5 |
| 3 | 0.100223 | 0 | 4.26 × 10-5 |
| 4 | 0.075527 | 0 | 3.02 × 10-4 |
| 5 | 0.048213 | 0 | 6.35 × 10-4 |
| 6 | 0.043825 | 0 | 1.32 × 10-3 |
| 7 | 0.070928 | 0 | 5.115 × 10-3 |
| 8 | 0.045950 | 0 | 0.025809 |
| 9 | 0.033264 | 0 | 0.083786 |
| 10 | 0.037807 | 0 | 0.480111 |
| 11 | 0.004711 | 0 | 1.55232 |
| 12 | 0.003099 | 0 | 2.2558 |
| 13 | 0.037882 | 0 | 25.5829 |
| 14 | 0.040110 | 0 | 10176.7 |
| 15 | 0.000997 | 0 | 277165 |
4. Joint Simulation of the Entire Wax Injection Molding Process
The dimensional accuracy and stability of the wax pattern are important factors affecting the quality of the final casting. To achieve precise control of the dimensional deformation of the parts in each process of precision casting, it is necessary to study the stress field and displacement field of the wax pattern forming process. The dimensional deformation of wax injection molding mainly consists of three parts: the shrinkage deformation of the wax pattern during cooling in the die, the stress/strain redistribution after demolding, and the viscoelastic deformation during storage. The traditional trial-and-error method has high cost and a long cycle, so it is necessary to use numerical simulation to quantitatively analyze and predict the dimensional deformation of the entire wax pattern preparation process.
4.1 Simulation Workflow
It is assumed that during the filling and packing process, the wax pattern is subjected to the packing pressure and is always completely attached to the cavity. After the packing process ends, the calculation of the shrinkage deformation of the wax pattern begins. Wax injection molding is a complex multi-physics coupling process, including mold flow analysis, temperature field analysis, stress field analysis, and displacement field analysis. To simplify the workload, I used both MoldFlow and Abaqus software to carry out the numerical simulation of the shrinkage deformation of the wax injection molding process. The filling and packing process of the wax pattern was simulated based on MoldFlow, and the initial stress field and initial temperature field obtained from the mold flow analysis of the wax pattern were imported into Abaqus as initial conditions to calculate the stress field and displacement field of the subsequent process.
The stress and deformation of the wax injection molding process are mainly divided into three stages. The first is the cooling shrinkage of the wax pattern in the die. Due to the constraint of the cavity on the wax pattern, the wax pattern cannot deform freely. The second is the stress release during demolding. The wax pattern deforms due to the uneven distribution of thermal residual stress/strain. The third is the shrinkage deformation during the post-demolding storage stage. The specific calculation flow is as follows. The first step is the simulation calculation of the wax pattern filling and packing process, which is carried out in the injection molding software MoldFlow. The second step is the calculation of the stress field and displacement field of the wax pattern continuing to cool in the die after the packing ends. The third step is the stress release at the moment the wax pattern is removed from the die. The deformation of the wax pattern during packing and in-mold cooling is constrained by the cavity or core, and the residual stress is released during demolding, and the stress and strain are redistributed. The fourth step is the shrinkage deformation of the wax pattern during storage outside the die. This stage can be divided into two processes. After demolding, the temperature of the wax pattern is still higher than room temperature, so it continues to cool to room temperature. After the wax pattern cools to room temperature, there is still a certain relaxation effect, and viscoelastic deformation continues until the dimensions stabilize. The second to fourth stages are carried out in the finite element software Abaqus.
4.2 Filling and Packing Simulation
The filling process is the initial step of the wax forming in the cavity. Because the filling process involves the non-Newtonian flow of the wax and unsteady heat transfer, the properties of the wax change with temperature, pressure, and shear rate, and the melt flow front is a free surface that changes with time, the filling process is a very complex process. The “filling + packing” analysis of MoldFlow can simulate the process of the wax from entering the cavity to filling the cavity and the subsequent packing and feeding.
A three-way pipe wax pattern was selected as the research object. Before simulation, the CAD solid file of the wax pattern was imported into MoldFlow, and then a 3D solid mesh was divided for the wax pattern. The injection process parameters are listed in Table 5.
| Parameter | Value |
|---|---|
| Wax injection temperature (°C) | 55 |
| Die temperature (°C) | 22 |
| Filling time (s) | 5.5 |
| Packing pressure (MPa) | 1.5 |
| Packing time (s) | 40 |
The filling process of the three-way pipe wax pattern showed that, due to the high viscosity of the wax, the wax filled uniformly from the center to the periphery, and finally completed filling at the short-axis pipe. The filling process was stable. When the wax fills the cavity, the wax near the die wall cools first and rapidly forms a solidified layer on the cavity surface, while the wax in the central layer cools more slowly and has a lower viscosity, maintaining a flowing state. The flowing melt in the central layer pushes the wax forward, causing the flow front to push the wax to both sides like a fountain; this is the “fountain effect”. The wax has a lower thermal conductivity than metal, and the heat transfer of the wax melt is slow. During filling, the temperature from the cavity surface to the central layer gradually increases, the wax viscosity gradually increases, and the flow gradually accelerates. The wax in the central layer flows in the direction opposite to the filling direction toward the cavity surface.
The outer side of the long-axis pipe of the three-way pipe wax pattern is thicker. The average temperature of the wax at the thick wall position is high, the viscosity is low, the flow resistance is small, the flow speed is fast, and a “race-track effect” is formed. Because the gate is located at the center, after the wax enters the cavity, it flows uniformly from the center to the periphery, and the filling on both sides of the thick wall is later. After a period, the flow speed of the wax at the thick wall positions on both sides is obviously accelerated, and finally the flow of the wax at the thick wall position is faster than that at the center.
The flow front temperature of the three-way pipe wax pattern and the pressure at the velocity/pressure switchover were obtained. The flow front temperature of the wax in the entire cavity was 54.35–55.00 °C, with a variation of only 0.65 °C, which meets the flow front variation requirements. At the velocity/pressure switchover, the maximum pressure was 0.0545 MPa, and the pressure at the short-axis pipe, which is the last area to be filled, was the smallest. To verify the accuracy of the wax flow state in the MoldFlow filling simulation, a non-center injection experiment was carried out on the three-way pipe wax pattern. The experimental results showed that the flow front of the wax at different times was basically consistent with the simulation results, and the flow front of the wax reflected the fountain effect.
4.3 In-Mold Cooling Analysis
Based on the residual stress and temperature field distribution at the end of packing obtained from MoldFlow, Abaqus was used to simulate the stress field and displacement field of the wax pattern during in-mold cooling. To avoid the difficulty of convergence caused by direct thermal-mechanical coupling and to improve the computational efficiency of the system, a temperature-stress sequential coupling method was adopted, that is, the temperature field of the wax pattern during the entire cooling process inside and outside the die was calculated first, and then the temperature result file obtained from the calculation was used as the input for the stress and deformation calculation flow.
Referring to the actual production injection process parameters, the cooling time of the three-way pipe wax pattern was set to 300 s. Because the thermal expansion coefficient of the metal die is much smaller than that of the wax, to simplify the calculation, the thermal expansion of the die on the wax pattern forming was not considered, the die was regarded as a rigid body, and the contact between the wax pattern and the die was defined as hard contact in the normal direction.
The residual stress and displacement fields of the three-way pipe wax pattern before demolding were obtained, and the elastic model and the viscoelastic model were compared. A considerable part of the thermal residual stress of the viscoelastic material relaxed during the cooling process, and the relaxation effect was very obvious. At the same time, due to creep, the deformation of the viscoelastic model was larger than that of the elastic model. However, because the deformation of the wax pattern was constrained by the cavity or core, the overall difference in deformation between the two models before the end of cooling was not large. The stress at the gate and at a point on the inner side of the pipe along the thickness direction decreased by about 30% during the die cooling process, showing obvious stress relaxation.
4.4 Demolding Stress Release Analysis
The dimensional deformation of the wax pattern preparation process can be divided into three parts. The first is the cooling shrinkage in the die. The wax pattern is constrained by the cavity or core during filling, packing, and in-mold cooling, and does not undergo obvious warpage. During this process, the temperature of the wax pattern drops significantly, and the dimensional deformation of the wax pattern in the die is mainly manifested as volume shrinkage. The second is the warpage deformation during demolding. When just demolded, the stress is released, and the wax pattern deforms due to thermal residual stress and uneven stress/strain distribution. The demolding stage lasts for a short time, and the wax pattern temperature remains basically unchanged. The third is the viscoelastic deformation during storage. Because the temperature of the wax pattern after demolding is still higher than room temperature, it continues to cool and deform. After cooling to room temperature, the wax pattern still has a stress relaxation effect and continues to deform at room temperature until the dimensions stabilize.
I took the state of the wax pattern at the end of in-mold cooling as the initial condition for the calculation of the demolding stress release. The effect of the ejection mechanism on the wax pattern was not considered, and it was assumed that all constraints of the die were instantaneously released, with a demolding time of 1 s. The deformation of the wax pattern during the demolding stage is unconstrained free deformation. Because the demolding time is short, the relaxation and creep effects are not obvious, and an elastic model was used to simplify the calculation.
During demolding, because the constraint of the die is removed, the stress field and displacement field of the wax pattern are redistributed. The thermal residual stress of the wax pattern is released in a very short time, and there is little change thereafter. The displacement field of the wax pattern during the demolding stage is also redistributed in a very short time, which is basically completed instantaneously. To compare the difference between the viscoelastic model and the elastic model in the warpage deformation after demolding, simulations were carried out separately. The three-way pipe wax pattern as a whole shrank toward the short-axis pipe side. This is because the intersection of the long-axis pipe and the short-axis pipe is a thick-wall area, where the wax cools slowly, the average temperature is high, and the shrinkage is large. Because the viscoelastic model has stress relaxation, a large part of the stress during in-mold cooling is relaxed, and the thermal residual stress before demolding is relatively small, so the warpage deformation after demolding is smaller than that of the elastic model. The maximum deformation of the elastic model was 1.527 mm, and the maximum deformation of the viscoelastic model was 1.279 mm. The deformation calculated by the elastic model was about 19% higher than that of the viscoelastic model.
4.5 Post-Demolding Storage Deformation
The demolded wax pattern is not directly used for slurry dipping and sand stuccoing to prepare the shell for subsequent processes. Instead, it is stored in a constant-temperature and constant-humidity environment for a period until its dimensions stabilize before use. It was assumed that the wax pattern is placed on a horizontal surface immediately after demolding.
The deformation of the wax pattern after standing horizontally for 24 h was calculated by numerical simulation. The three-way pipe wax pattern had a tendency to shrink and deform toward the short-axis pipe. After demolding, the long-axis pipe of the three-way pipe wax pattern was placed horizontally facing downward. Key points were marked on the wax pattern to analyze the dimensional changes. The key dimension numbers and descriptions of the three-way pipe wax pattern are listed in Table 6.
| Number | Description |
|---|---|
| $X_A$ | Displacement difference between $A_1$ and $A_2$ in the X-axis direction |
| $Y_A$ | Displacement difference between $A_1$ and $A_2$ in the Y-axis direction |
| $Z_B$ | Displacement difference between $B_1$ and $B_2$ in the Z-axis direction |
| $X_C$ | Displacement difference between $C_1$ and $C_2$ in the X-axis direction |
| $Z_D$ | Displacement difference between $D_1$ and $D_2$ in the Z-axis direction |
| $X_{E1}$ | Displacement difference between $A_1$ and $E_1$ in the X-axis direction |
| $X_{E2}$ | Displacement difference between $A_2$ and $E_2$ in the X-axis direction |
| $Y_{A12}$ | Distance between $A_1$ and $A_2$ in the Y-axis direction |
| $Z_{B12}$ | Distance between $B_1$ and $B_2$ in the Z-axis direction |
| $X_{C12}$ | Distance between $C_1$ and $C_2$ in the X-axis direction |
| $Z_{D12}$ | Distance between $D_1$ and $D_2$ in the Z-axis direction |
| $X_{AE1}$ | Distance between $A_1$ and $E_1$ in the X-axis direction |
| $X_{AE2}$ | Distance between $A_2$ and $E_2$ in the X-axis direction |
Because the thick-wall position has a large shrinkage after cooling, the uneven shrinkage causes the three-way pipe wax pattern to have a tendency to warp toward the short-axis pipe after demolding. When the wax pattern is placed on a horizontal surface, its top surface is not horizontal. The stress and strain of point $A_1$ and point $A_2$ after standing horizontally for 24 h show that the stress and strain of point $A_1$ are smaller than those of point $A_2$, and the wax pattern still has stress relaxation and creep after cooling to room temperature. Table 7 shows the comparison between the experimental measurement results and the simulation results of the characteristic dimensions of the three-way pipe wax pattern. The overall shrinkage of the three-way pipe wax pattern was about 2 mm, and the wax pattern dimensions calculated by the simulation were generally smaller, but the overall deviation was controlled within 0.3 mm, indicating that the calculation results are relatively accurate.
| Characteristic dimension | $X_{AE1}$/mm | $X_{AE2}$/mm | $Y_{A12}$/mm | $Z_{B12}$/mm | $X_{C12}$/mm | $Z_{D12}$/mm |
|---|---|---|---|---|---|---|
| Die dimension | 170.5 | 170.5 | 117 | 117 | 95 | 95 |
| Measured dimension | 168.62 | 168.25 | 115.53 | 115.28 | 93.65 | 93.67 |
| Simulated dimension | 168.50 | 168.15 | 115.29 | 114.98 | 93.49 | 93.54 |
| Deviation | 0.12 | 0.10 | 0.24 | 0.30 | 0.16 | 0.13 |
4.6 Dimensional Deformation Law
The preparation of the wax pattern is the first process of precision casting. The dimensional deformation of the wax pattern forming process is inherited by subsequent processes. The dimensional instability of the wax pattern causes casting dimensional errors, and the quality of the wax pattern has the most significant influence on the dimensional accuracy of the casting. The dimensional errors of wax pattern forming mainly come from die design errors, in-mold pressing errors, and out-of-die warpage deformation errors. Among them, die design errors are systematic errors that can be avoided before production. In-mold pressing errors refer to the cooling shrinkage of the wax pattern in the die, which can be controlled by designing reasonable wax injection process parameters. Out-of-die warpage deformation is divided into two stages: the stress/strain redistribution after demolding, and the viscoelastic deformation during storage.
The deformation of the three-way pipe wax pattern was mainly concentrated within 9 h after demolding, at which time the deformation of the wax pattern accounted for more than 98% of the total deformation. The deformation proportions of the wax pattern in the in-mold cooling, demolding, and storage stages were calculated to be about 35%, 40%, and 25%, respectively. During the demolding stage, the stress/strain of the wax pattern is redistributed, and the key dimensions change abruptly. After demolding, the key dimension $X_{E2}$ first rises slowly, then decreases, and gradually stabilizes. This is because the center of gravity of the three-way pipe wax pattern during the storage stage is not located at the geometric center but is biased toward the short-axis pipe, and the distance between $A_2$ and $E_2$ is elongated. Because the wax pattern is still higher than room temperature after demolding, the wax pattern undergoes a period of cooling shrinkage and then continues to deform at room temperature, so the distance between $A_2$ and $E_2$ undergoes a process of first decreasing and then increasing after demolding.
The displacement difference between $A_1$ and $A_2$ in the X-axis direction can characterize the warpage deformation degree of the three-way pipe wax pattern. When the wax pattern is cooled in the die, due to the constraint of the cavity or core, the $X_{AE1}$ direction cannot shrink freely. In the die, the shrinkage of point $A_2$ in the X direction is greater than that of point $A_1$, so $X_A$ is initially negative and gradually decreases. The intersection of the long-axis pipe and the short-axis pipe of the wax pattern cannot deform due to the constraint of the die, resulting in greater residual stress, and the residual stress at the intersection is obviously greater than that at the side. During demolding, the stress is released, the deformation at the intersection is large, and point $A_1$ shrinks sharply inward. During the storage stage, the wax pattern continues to undergo viscoelastic deformation, and the height difference between $A_1$ and $A_2$, that is, $X_A$, continues to increase. The warpage deformation degree of the wax pattern gradually increases and finally stabilizes.
5. Influence of Process Parameters on Dimensional Deformation
The wax injection process parameters have a significant influence on the dimensional accuracy of the wax pattern. To obtain a high-quality wax pattern, it is necessary to deeply study the influence law of process parameters on wax pattern preparation and strictly control the process parameters of the wax pattern forming process. Traditional wax pattern preparation mostly uses the trial-and-error method, which requires repeated modification of process parameters to ensure wax pattern quality, resulting in a long trial production cycle and high cost. There are many wax injection process parameters that need to be adjusted in production, and the influence of each parameter on the forming quality of the wax pattern is different, and there are interactions between different parameters.
5.1 Selection of Process Parameter Range
The process parameters of the three-way pipe wax pattern were studied. First, the range of wax injection process parameters was selected in combination with actual production experience. Second, the Taguchi method was used to screen out the process parameters that have little influence on the dimensional deformation of the wax pattern. Then, the response surface of the wax pattern forming quality and process parameters was fitted. Finally, the multi-objective optimization of the wax pattern forming quality was realized based on a genetic algorithm. The analysis and range selection of the wax injection process parameters are as follows.
The K512 wax solidifies below 49 °C. If the wax injection temperature is too low, the wax will be difficult to fill the cavity, and even short shots may occur. The thermal conductivity of the wax is poor, and an increase in the wax injection temperature makes the wax pattern need a longer time to cool, prolonging the forming cycle of wax pattern pressing. In addition, a large temperature difference easily forms a larger shrinkage deformation and affects the dimensional accuracy of the subsequent shell. According to actual production experience, the wax injection temperature range was selected as 55–65 °C.
The die temperature determines the cooling rate of the wax pattern. A higher die temperature can promote uniform cooling of the wax pattern, but it also increases the volume shrinkage of the wax pattern after demolding. To effectively control the forming quality of the wax pattern and the dimensional stability during the pressing process, it is necessary to study the influence of the die temperature on the shrinkage deformation of the wax injection molding process. When the die temperature is too high, the strength of the wax pattern after demolding is low, and a large shrinkage deformation will occur after further cooling to room temperature. Therefore, the die temperature should not be set too high, so as to avoid large shrinkage deformation of the wax pattern and a long forming cycle, nor should it be set too low, which would make the wax pattern difficult to fill. According to actual production experience, the die temperature range was selected as 19–25 °C.
Generally speaking, the injection machine mainly adopts speed control during the wax filling stage, and switches to pressure control when filling is about to end. At this time, the injection machine still applies a certain pressure to the wax to achieve the feeding effect. The filling time is the time for the wax to fill the cavity, reflecting the injection speed of the wax. The shorter the filling time, the higher the injection speed. If the filling is too fast, the wax will become turbulent, or even splash. As the filling time increases, the injection speed decreases. When the wax has not completely filled the cavity, part of the wax has cooled and solidified, hindering the subsequent injection of the wax. To ensure the forming quality of the wax pattern, the filling time should not be too small or too large. According to actual production experience, the injection time range of the three-way pipe wax pattern was selected as 4–7 s.
When the wax just begins to be injected, the wax can fill the cavity at a certain injection speed under a relatively low injection pressure. When the wax fills the cavity, the role of the injection pressure is to compact the wax in the die. At this time, the injection pressure is the packing pressure, and the injection machine also switches from speed control to pressure control. If the packing pressure is too low, short shots and shrinkage of the wax will occur. However, an excessively large packing pressure will make demolding difficult and easily produce flashes on the wax pattern, affecting the surface quality. Therefore, the selection of packing pressure requires consideration of many aspects. The packing pressure range was selected as 1.5–3.5 MPa.
The length of the packing time also affects the forming quality of the wax pattern. Prolonging the packing time can improve the dimensional stability of the wax pattern and obtain a dense product, but it increases the difficulty of demolding. Moreover, when the gate solidifies, extending the packing time is basically meaningless. According to actual production experience, the packing time range was selected as 20–60 s.
5.2 Taguchi Method for Significance Analysis
The traditional research on the influence of injection molding process parameters mostly uses the univariate analysis method, in which only one variable changes at a time and other variables remain constant. This method is difficult to evaluate the degree of influence of different process parameters on the forming quality of the wax pattern, and cannot separate the main effect from the interaction effect. If a full factorial experimental design is used, for the five process parameters selected above, if each parameter is set at three levels, 243 experiments are required. The number of experiments required is large, consuming a lot of manpower, material resources, and time. To evaluate the degree of influence of each process parameter on the forming quality of the wax pattern, effectively screen out the parameters with significant influence, and narrow the selection range for subsequent optimization design, other design of experiment methods are needed.
The Taguchi method is a robust parameter design method that evaluates product quality through the signal-to-noise ratio, while considering the influence of controllable factors and noise on the response, and seeks the combination of controllable factors with the strongest resistance to noise. It is widely used in product design and development. I studied the influence degree of wax injection process parameters on the forming quality of the wax pattern based on the Taguchi method and screened out the factors with significant influence for response surface experimental design.
I selected five controllable process parameters: die temperature (A), packing pressure (B), filling time (C), wax injection temperature (D), and packing time (E). Based on the parameter selection range discussed in the previous section, the Taguchi method was used to design the experiment. The average volume shrinkage and maximum deformation of the three-way pipe wax pattern after demolding without constraint were taken as the test indicators to comprehensively evaluate the shrinkage deformation of the wax injection molding. The mean value and signal-to-noise ratio were used to measure the influence degree of each process parameter on the dimensional deformation of the wax pattern pressing process. The test factors and levels are listed in Table 8, and the experimental design and results are listed in Table 9.
| Test level | A/°C | B/MPa | C/s | D/°C | E/s |
|---|---|---|---|---|---|
| 1 | 19 | 1.5 | 4 | 55 | 20 |
| 2 | 25 | 3.5 | 7 | 65 | 40 |
| No. | A | B | C | D | E | Shrinkage/% | Deformation/mm |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 | 1 | 6.5056 | 1.2923 |
| 2 | 1 | 1 | 1 | 2 | 2 | 7.2122 | 1.3189 |
| 3 | 1 | 2 | 2 | 1 | 1 | 6.2556 | 1.2522 |
| 4 | 1 | 2 | 2 | 2 | 2 | 6.6905 | 1.2833 |
| 5 | 2 | 1 | 2 | 1 | 2 | 7.0792 | 1.2970 |
| 6 | 2 | 1 | 2 | 2 | 1 | 8.2306 | 1.3483 |
| 7 | 2 | 2 | 1 | 1 | 2 | 6.9728 | 1.2569 |
| 8 | 2 | 2 | 1 | 2 | 1 | 8.2096 | 1.3509 |
The signal-to-noise ratio is calculated as follows, and since the test quality indicators are all smaller-the-better characteristics, the signal-to-noise ratio is:
$$S/N = -10 \times \log\left(\frac{1}{n}\sum_{i=1}^{n} y_i^2\right)$$
where $S/N$ is the signal-to-noise ratio, $y_i$ is the evaluation index, and $n$ is the number of tests. For smaller-the-better characteristics, a smaller signal-to-noise ratio is better. The mean value response and signal-to-noise ratio response of the shrinkage are listed in Table 10. For the volume shrinkage of the wax pattern after forming, the mean value and signal-to-noise ratio evaluation of the significance of the process parameters gave basically the same results, and both reflected the same order of influence. The order of influence significance from large to small is: die temperature, wax injection temperature, packing time, packing pressure, and filling time. The filling time is the process parameter with the smallest influence on the volume shrinkage of the wax pattern after pressing.
| Level | A | B | C | D | E |
|---|---|---|---|---|---|
| 1 | 6.666 | 7.257 | 7.225 | 6.703 | 7.300 |
| 2 | 7.623 | 7.032 | 7.064 | 7.586 | 6.989 |
| Delta | 0.957 | 0.225 | 0.161 | 0.882 | 0.312 |
| Rank | 1 | 4 | 5 | 2 | 3 |
For the deformation of the wax pattern, the filling time is also the process parameter with the smallest influence. The order of influence of the other process parameters on the deformation from large to small is: wax injection temperature, packing pressure, die temperature, and packing time. The packing pressure has a very significant influence on the warpage deformation of the wax pattern after demolding. If the packing pressure is too low, the compression of the wax will be insufficient, and the volume shrinkage of the wax pattern after cooling will be too large. If the packing pressure is too high, demolding will be difficult, and die swelling may be caused in severe cases. Therefore, for the indicator of deformation after demolding, the wax injection temperature and packing pressure should be mainly controlled.
Through the Taguchi experimental method, it was found that the filling time has little influence on the shrinkage deformation of the wax pattern during pressing. To further study the influence of filling time on the forming quality of the wax pattern, according to actual production conditions, eight groups of simulations were designed with filling times within 2–9 s, with a wax injection temperature of 60 °C, a die temperature of 20 °C, a packing pressure of 2 MPa, and a packing time of 40 s. After packing, the wax pattern was cooled in the die for 300 s and then freely deformed without constraint after demolding. The maximum clamping force, maximum sink mark depth, average volume shrinkage, and maximum warpage deformation in the injection molding simulation were used as four indicators to comprehensively evaluate the forming process conditions and forming quality of the wax pattern. The design range of the filling time and the corresponding forming quality are listed in Table 11.
| Filling time/s | Clamping force/t | Sink mark depth/mm | Shrinkage/% | Deformation/mm |
|---|---|---|---|---|
| 2 | 2.36 | 0.7374 | 6.7426 | 1.2936 |
| 3 | 4.89 | 0.7702 | 6.9522 | 1.2975 |
| 4 | 2.61 | 0.7433 | 6.7334 | 1.2955 |
| 5 | 3.40 | 0.7379 | 6.6977 | 1.2973 |
| 6 | 4.84 | 0.7592 | 6.8992 | 1.2979 |
| 7 | 5.98 | 0.7651 | 6.8919 | 1.2966 |
| 8 | 7.68 | 0.8203 | 6.9193 | 1.2973 |
| 9 | 7.69 | 0.7940 | 6.9158 | 1.2966 |
With the increase in filling time, the sink mark depth first decreases and then increases, and the results for 4–7 s are relatively small. The volume shrinkage first decreases and then increases with the filling time. This is because the cooling process of the wax pattern has actually already begun in the filling stage. An increase in filling time means that the cooling time of the wax pattern in the die is lengthened, and the volume shrinkage at demolding is relatively large. The warpage deformation of the wax pattern at demolding has no obvious relationship with the filling time. When the injection speed is slow, part of the wax has cooled and solidified before the cavity is completely filled, hindering the subsequent injection of the wax, so the clamping force generally increases with the increase in filling time.
It can be seen that the filling time has little influence on the shrinkage deformation of the wax pattern after forming. To narrow the range of factors considered to reduce the number of experiments, and at the same time select the process parameters with the most significant influence on the shrinkage deformation of the wax pattern during pressing to achieve optimization analysis, the filling time was kept constant, and the influence of the four process parameters of wax injection temperature, die temperature, packing pressure, and packing time on the dimensional deformation of the wax pattern was analyzed. To ensure the forming quality of the wax pattern, the filling time should not be too small or too large. According to actual production experience, the filling time of the three-way pipe wax pattern was selected as 6 s.
5.3 Response Surface Methodology Optimization Analysis
Response surface methodology is a test design and optimization method. By carrying out sampling experiments with local representative sample points, the functional relationship between factors and fitted values is mathematically modeled, and the quadratic regression equation between independent variables and response values is fitted. On this basis, the main effect of factors and the interaction between factors can be studied, the response values outside the sample points can be predicted, and the analysis and solution of the optimal process parameter combination can be realized.
Based on the analysis results of the Taguchi method, the response surface methodology was used to study the influence of the four process parameters of wax injection temperature, die temperature, packing pressure, and packing time on the dimensional deformation of the wax pattern during pressing. The volume shrinkage and deformation of the wax pattern after demolding were used as measurement indicators, and the analytical expressions between factors and response values were fitted to estimate the main effect of each factor and the magnitude of the interaction effects between factors at all levels. A central composite face-centered design was used, and the filling time was screened out. The factor levels are listed in Table 12, and the experimental design and simulation results are listed in Table 13.
| Level | A/°C | B/MPa | C/°C | D/s |
|---|---|---|---|---|
| -1 | 19 | 1.5 | 55 | 20 |
| 0 | 22 | 2.5 | 60 | 40 |
| 1 | 25 | 3.5 | 65 | 60 |
| No. | A | B | C | D | Shrinkage/% | Deformation/mm |
|---|---|---|---|---|---|---|
| 1 | -1 | -1 | -1 | -1 | 6.4082 | 1.2919 |
| 2 | 1 | -1 | -1 | -1 | 7.1877 | 1.2980 |
| 3 | -1 | 1 | -1 | -1 | 6.1319 | 1.2472 |
| 4 | 1 | 1 | -1 | -1 | 7.0338 | 1.2574 |
| 5 | -1 | -1 | 1 | -1 | 7.6278 | 1.3364 |
| 6 | 1 | -1 | 1 | -1 | 8.2349 | 1.3483 |
| 7 | -1 | 1 | 1 | -1 | 7.6117 | 1.3245 |
| 8 | 1 | 1 | 1 | -1 | 8.2136 | 1.3483 |
| 9 | -1 | -1 | -1 | 1 | 6.4074 | 1.2919 |
| 10 | 1 | -1 | -1 | 1 | 7.1381 | 1.2976 |
| 11 | -1 | 1 | -1 | 1 | 6.1110 | 1.2468 |
| 12 | 1 | 1 | -1 | 1 | 7.0124 | 1.2561 |
| 13 | -1 | -1 | 1 | 1 | 7.1599 | 1.3183 |
| 14 | 1 | -1 | 1 | 1 | 7.7783 | 1.3208 |
| 15 | -1 | 1 | 1 | 1 | 6.8674 | 1.2866 |
| 16 | 1 | 1 | 1 | 1 | 7.3914 | 1.2889 |
| 17 | -1 | 0 | 0 | 0 | 6.7102 | 1.2870 |
| 18 | 1 | 0 | 0 | 0 | 7.3259 | 1.2906 |
| 19 | 0 | -1 | 0 | 0 | 7.1100 | 1.3063 |
| 20 | 0 | 1 | 0 | 0 | 6.7923 | 1.2681 |
| 21 | 0 | 0 | -1 | 0 | 6.8632 | 1.2804 |
| 22 | 0 | 0 | 1 | 0 | 7.3625 | 1.3052 |
| 23 | 0 | 0 | 0 | -1 | 7.4542 | 1.3030 |
| 24 | 0 | 0 | 0 | 1 | 6.9991 | 1.2885 |
| 25 | 0 | 0 | 0 | 0 | 6.9991 | 1.2885 |
The classical response surface model is based on the assumption of homogeneity of variance. A first-order model is first fitted to determine whether the current independent variable factor level is close to the optimum position of the response surface. The first-order regression equation takes the following form:
$$y = a_0 + \sum_{i=1}^{n} a_i x_i + \varepsilon$$
where $y$ is the response value, $a_0$ is a constant, $x_i$ is the $i$-th component of the $n$-dimensional independent variable factor, $a_i$ is the linear effect of the factor $x_i$, and $\varepsilon$ is the error. If the response surface contains curvature, a higher-order polynomial model should be used. The second-order regression equation has the following form:
$$y = a_0 + \sum_{i=1}^{n} a_i x_i + \sum_{i<j}^{n} +=""
where $a_{ij}$ is the interaction effect between factors $x_i$ and $x_j$, and $a_{ii}$ is the quadratic effect of factor $x_i$. For convenience of discussion, the average volume shrinkage of the wax pattern is denoted by the symbol $\varphi$, and the deformation is denoted by the symbol $\delta$, with units of % and mm, respectively.
Considering the interaction between factors, a complete quadratic regression equation was fitted. The terms considered include A (die temperature), B (packing pressure), C (wax injection temperature), D (packing time), the square terms of each factor, and the two-factor interaction terms. The analysis of variance for the volume shrinkage of the wax pattern showed that the P values of C, A, D, CD, B, and DD were all less than 0.05, indicating that these terms significantly affect the volume shrinkage of the wax pattern. The adjusted coefficient of determination was greater than 95%, proving the accuracy of the response surface model fitting. The quadratic term was statistically significant, proving that there is curvature in the response surface, that is, the influence of packing pressure on shrinkage is not a monotonic linear relationship.
After removing the terms with insignificant influence on shrinkage, the regression equation for the volume shrinkage is:
$$\varphi = -3.314 + 0.11631 A – 0.1048 B + 0.1483 C + 0.0554 D + 0.000326 D^2 – 0.001499 C \cdot D$$
The P values of all terms in the new model are less than 0.05, and the influence of each factor on the wax pattern shrinkage is significant. The predicted coefficient of determination is above 0.9, indicating that the model has a certain degree of accuracy and good predictive ability.
Using the same method to analyze the influence of each factor and the interaction between factors on the deformation of the wax pattern, the analysis of variance showed that the order of influence significance from large to small is: C (wax injection temperature), B (packing pressure), D (packing time), CD (interaction between wax injection temperature and packing time), BC (interaction between packing pressure and wax injection temperature), and A (die temperature). The other factors are not statistically significant for the deformation of the wax pattern. After removing the terms with insignificant influence on deformation, the regression equation for the deformation is:
$$\delta = 1.0185 + 0.001394 A – 0.0883 B + 0.00505 C + 0.00484 D + 0.001207 B \cdot C – 0.000088 C \cdot D$$
For this model, the P values of all terms are less than 0.05, the adjusted coefficient of determination is greater than 90%, and the response surface model fits well. The predicted coefficient of determination is close to 90%, indicating that the regression equation can predict the response values outside the measured data set relatively accurately, and the fitting of the actual deformation is good.
5.4 Influence Law Analysis
To further study the influence of wax injection process parameters on the dimensional deformation of the wax pattern, two of the process parameters were kept unchanged, and a three-dimensional response surface was drawn based on the response surface equation before removing the insignificant terms to represent the influence of the other two factors on the free shrinkage deformation of the wax pattern after demolding, and the main effect of each factor and the interaction effect between factors on the forming quality of the wax pattern were analyzed.
For the single-factor influence, the wax injection temperature and die temperature have significant effects on the average volume shrinkage and deformation of the wax pattern. Reducing the temperature can effectively reduce the dimensional deformation of the wax pattern after demolding. Although reducing the wax injection temperature and die temperature will increase the viscosity of the wax during filling to a certain extent and cause a decrease in the filling ability of the melt, a lower temperature can effectively shorten the cooling cycle of the wax pattern and enhance the dimensional stability of the wax pattern forming. Generally speaking, reducing the wax injection temperature and die temperature can improve the dimensional accuracy of the wax pattern, but if the temperature is too low, the flow resistance of the wax during filling is too large, which is not conducive to feeding, and short shots may even occur. The shrinkage deformation of the wax pattern is an important indicator of the forming quality of the wax pattern, but it is not the only indicator. Actual production should comprehensively consider various quality indicators of the wax pattern and balance the selection of appropriate process parameters.
Increasing the packing pressure can reduce the volume shrinkage and warpage deformation of the wax pattern. At the beginning of lengthening the packing time, the dimensional stability of the wax pattern can be improved, and the shrinkage and deformation of the wax pattern can be reduced. However, when the gate solidifies, extending the packing time is basically meaningless. This is also consistent with the trend of the curves.
For the interaction effect, the quadratic term of packing time is statistically significant, indicating that the influence of packing time on the volume shrinkage of the wax pattern is not simply monotonic. When the wax pattern gate solidifies, continuing to apply packing pressure no longer works. The volume shrinkage of the wax pattern decreases with the decrease in wax injection temperature. The influence of packing time on shrinkage has a transition point. At the beginning of extending the packing time, the volume shrinkage of the wax pattern decreases accordingly, but continuing to extend the packing time has little effect on the shrinkage. At the same time, increasing the wax injection temperature will correspondingly increase the transition point of the influence of packing time on the volume shrinkage of the wax pattern. At this time, the wax pattern gate solidifies later, and the packing pressure can act on the wax pattern for a longer time. The interaction between wax injection temperature and packing time is relatively significant. The interaction between packing pressure and wax injection temperature and the interaction between wax injection temperature and packing pressure have a relatively significant influence on the deformation of the wax pattern.
5.5 Multi-Objective Optimization Based on Genetic Algorithm
The quadratic regression equations of the volume shrinkage and deformation of the wax pattern were fitted based on the response surface methodology. To obtain the optimal process parameter combination for the shrinkage deformation of the wax pattern during pressing, it is necessary to comprehensively consider the effects of volume shrinkage and deformation so that these two indicators are minimized as a whole, thereby obtaining the optimal process parameters. To achieve this goal, I used a genetic algorithm to optimize the shrinkage deformation of the wax pattern during pressing.
A genetic algorithm is an optimization search algorithm based on biological evolution theory. Compared with traditional optimization methods such as enumeration and heuristics, the genetic algorithm has good convergence and robustness, high computational efficiency, and can avoid falling into a local optimum, and performs very well in dealing with complex problems.
The influence law of each factor on different response indicators is different. For volume shrinkage, the order of influence of each factor from large to small is wax injection temperature, die temperature, packing time, and packing pressure. For deformation, the order of influence of each factor from large to small is wax injection temperature, packing pressure, packing time, and die temperature. At the same time, the interaction of some factors also affects the shrinkage deformation of the wax pattern forming process. Therefore, it is unreasonable to seek the optimal process parameters for the wax pattern pressing process only for one evaluation indicator. It is necessary to simultaneously optimize the two objectives of volume shrinkage and deformation of the wax pattern, which is actually a multi-objective optimization problem.
To solve the multi-objective optimization problem of wax pattern forming quality, the actual engineering problem can be transformed into a mathematical model, expressed as follows:
$$
\begin{cases}
\min F(x) = [f_1(x), f_2(x), \cdots, f_k(x)] \\
\text{s.t.} \begin{cases}
g_i(x) \leq 0, \quad i = 1, 2, \cdots, m \\
h_i(x) = 0, \quad i = 1, 2, \cdots, n \\
a_i \leq x_i \leq b_i, \quad i = 1, 2, \cdots, p
\end{cases}
\end{cases}
$$
where $x$ is the independent variable, $f_k(x)$ is the $k$-th objective function, $g_i(x)$ is the inequality constraint, $h_i(x)$ is the equality constraint, $a_i$ and $b_i$ are the minimum and maximum values of the $i$-th design variable in $X = [x_1, x_2, \cdots, x_p]^T$, and $p$ is the number of design variables.
In actual engineering practice, different objective values often show different growth or decay patterns, and there may be contradictions between objectives. Simultaneously achieving the optimal state of all objectives is usually unrealistic. Therefore, the purpose of multi-objective optimization is not to find a single optimal solution for all objective functions, but to find an optimal solution set that satisfies all objective functions as much as possible, and to select an appropriate parameter combination from this solution set according to the actual production state.
The specific algorithm used was the NSGA-II multi-objective genetic algorithm. This algorithm is improved on the basis of the first-generation non-dominated sorting genetic algorithm. It adopts a fast non-dominated sorting algorithm, uses crowding distance and a crowding comparison operator, and introduces an elite automatic retention strategy, which reduces the complexity of the non-dominated sorting genetic algorithm and has the advantages of high computational efficiency and good convergence. The multi-objective genetic algorithm selects and sorts individuals according to the non-dominated rank value and crowding degree, then selects individuals, while the traditional genetic algorithm directly selects individuals through fitness. The parent generation generates the offspring through crossover and mutation, and suitable individuals in the combined set of parent and offspring are screened out through non-dominated sorting, and then individuals in the unified rank layer are selected through crowding distance sorting to obtain a new population set.
After the response surface model was established, the genetic algorithm was used to optimize according to the constraint conditions and optimization objectives of the response. Taking the four process parameters as constraints, the die temperature, packing pressure, wax injection temperature, and packing time as optimization variables, and the comprehensive index of dimensional deformation of the wax pattern pressing process as the optimization objective, the genetic algorithm was used to find the minimum value of the comprehensive index within the constraint domain. The expression is as follows:
$$
\begin{cases}
\text{Optimization variables: } X = (A, B, C, D) \\
\text{Constraints: } \begin{cases}
19 \leq F_1(X) \leq 25 \\
1.5 \leq F_2(X) \leq 3.5 \\
55 \leq F_3(X) \leq 65 \\
20 \leq F_4(X) \leq 60
\end{cases} \\
\text{Optimization objective: } \text{Minimize}(\varphi, \delta)
\end{cases}
$$
where $\varphi$ and $\delta$ are the objective functions of the volume shrinkage and deformation of the wax pattern, respectively, and A, B, C, and D are the die temperature, packing pressure, wax injection temperature, and packing time, respectively. The program was written in a numerical computing environment, and the multi-objective optimization function was called in the main program. The Pareto optimization solution set was obtained after running.
From the Pareto curve, it can be seen that the volume shrinkage and deformation of the wax pattern are mutually restrictive. Therefore, how to select the optimal process parameters should be determined according to actual production requirements. The deformation in the Pareto curve changes very little, basically about 1.2460 mm. Therefore, under the premise that the deformation difference is not large, the shrinkage is preferentially selected. Considering comprehensively, the highest point of the Pareto curve was selected as the target value. The volume shrinkage of this target value was 6.1248%, and the deformation was 1.246 mm. The corresponding process parameter combination was a die temperature of 19 °C, a packing pressure of 3.5 MPa, a wax injection temperature of 55 °C, and a packing time of 41.5 s.
Table 14 shows the simulation results of the wax pattern under these process parameters. It can be seen that the relative error between the results predicted based on the response surface methodology and the genetic algorithm and the simulation results is small, proving that this method is reliable. The optimal process parameter combination not only reduces the volume shrinkage and deformation of the wax pattern but also provides a scientific basis for the process design and optimization of wax injection molding in precision casting.
| Item | $\varphi$/% | $\delta$/mm |
|---|---|---|
| Numerical simulation result | 6.1661 | 1.264 |
| Genetic algorithm optimization | 6.1248 | 1.246 |
| Error | 1.33% | 1.42% |
6. Conclusions and Future Work
Wax pattern preparation is the first process of precision casting and directly determines the dimensional shape of the shell. The dimensional error of the wax pattern accumulates and transfers to the next generation as the precision casting process advances, significantly affecting the surface quality and dimensional accuracy of the casting. I combined numerical simulation and experimental research to realize the numerical simulation of the stress and deformation of the entire wax pattern preparation process, explored the dimensional deformation law of the wax pattern, and optimized the wax injection process parameters. The main conclusions are as follows.
For the K512 wax used in precision casting, I selected the Cross-WLF model and the generalized Maxwell model as the viscosity constitutive model and the viscoelastic constitutive model of the wax. The glass transition temperature, linear viscoelastic region, and dynamic shear relaxation modulus of the wax were measured by a rotational rheometer. The Arrhenius model was used to fit the shift factor, and the relaxation modulus master curve was obtained by shifting and superposing. The relaxation time spectrum of the wax was fitted and normalized, and a three-dimensional viscoelastic deformation calculation model of the wax pattern was established. Based on the results of specific heat capacity measurement, thermal conductivity measurement, PVT property measurement, and rheological property measurement, a performance parameter database of the K512 wax was constructed through formula derivation.
The filling and packing process of the three-way pipe wax pattern was analyzed by MoldFlow, and the accuracy of the flow state simulation was verified by injection experiments. Abaqus was used to calculate the stress and deformation of the wax pattern preparation process, including the shrinkage deformation of the wax pattern during in-mold cooling, the stress/strain redistribution during demolding, and the viscoelastic deformation during the storage stage. The comparison of the measured results of key dimensions proved the accuracy of the simulation results, and the dimensional prediction deviation was within 0.3 mm. At the same time, the dimensional deformation law of the wax pattern preparation process was analyzed, and it was found that more than 98% of the deformation of the wax pattern was concentrated within 9 h after demolding.
For the three-way pipe wax pattern, the influence of wax injection process parameters on the forming quality of the wax pattern was explored, and the process parameter optimization was realized. Taking the volume shrinkage and deformation of the wax pattern as the evaluation indicators, and combining actual production experience to select the appropriate range of wax injection process parameters, the Taguchi method was used to analyze the influence degree of process parameters on the forming quality of the wax pattern. It was found that the filling time has the smallest influence on the volume shrinkage and deformation of the wax pattern. Therefore, the filling time was screened out, and the quadratic regression equations of the volume shrinkage and deformation of the wax pattern were fitted based on the response surface methodology, and the influence of process parameters on the forming quality of the wax pattern and the interaction between parameters were analyzed. Finally, the multi-objective optimization of the wax pattern forming quality was realized based on the genetic algorithm. The optimal wax injection process parameter combination was found to be a die temperature of 19 °C, a wax injection temperature of 55 °C, a packing pressure of 3.5 MPa, and a packing time of 41.5 s.
For future work, the viscoelastic deformation of the wax pattern during the storage stage was calculated by numerical simulation without considering complex restrictive conditions. In subsequent research, large and complex wax pattern parts should be selected as research objects to further analyze the influence law of process parameters and storage methods on the viscoelastic deformation of complex-structure wax patterns after demolding. In addition, the numerical simulation of the stress and deformation of the entire wax injection molding process was carried out, but a dimensional deformation control model of the wax pattern forming process was not established. In the future, the deformation mechanism of the wax pattern preparation process can be studied more deeply, and a dimensional error control model for the entire wax injection molding process can be established, so as to realize the quantitative prediction and precise control of the wax pattern forming dimensions through reverse deformation design. This will further promote the application of numerical simulation technology in precision casting and provide more powerful support for the manufacture of high-quality precision castings.