I have studied the entire stress and deformation history of wax injection molding for investment casting because the wax pattern is the first dimensional link in a long process chain. In investment casting, every dimensional error generated during wax injection can be inherited by the shell, then by the metal casting, and finally by the finished component. This inheritance is especially important in aerospace, automotive, medical, and energy applications, where thin walls, complex geometry, and high dimensional accuracy are required. I also note that the same logic appears in evaporative pattern casting, where a sacrificial pattern must retain its shape before and during thermal degradation. In evaporative pattern casting, pattern stability controls mold filling and final tolerance, just as wax pattern stability controls shell dimensions and final casting accuracy. Therefore, the numerical framework I developed for wax injection molding can also inform evaporative pattern casting when the pattern material is polymeric, foam-based, or otherwise thermally sensitive.
My central motivation was that wax preparation is not a single event. It includes filling, packing, in-mold cooling, demolding, and post-demolding storage. Many previous studies have focused on only one or two of these stages. I found that such a partial view is insufficient because the viscoelastic nature of wax means that stress relaxation and creep continue after the mold is opened. In evaporative pattern casting, a similar time-dependent response can occur during pattern aging or handling, and the same modeling philosophy is useful. I therefore constructed a full-process simulation route that combines mold-filling analysis with finite element stress analysis. The goal was not only to predict final dimensions but also to explain when and why the largest deformation occurs.
The wax I selected was a commercial low-shrinkage investment casting wax. I measured its thermal properties, rheological behavior, pressure-volume-temperature relation, Poisson ratio, and linear viscoelastic response. I then fitted a Cross-WLF viscosity model and a generalized Maxwell viscoelastic model. The Cross-WLF model was used for the filling and packing stages, while the generalized Maxwell model was used for in-mold cooling, demolding, and storage. This combination allowed me to represent both the fluid-like behavior during injection and the solid-like viscoelastic behavior after solidification. I also applied the time-temperature superposition principle so that dynamic modulus data measured at several temperatures could be shifted into a master curve. This master curve was then converted into a relaxation time spectrum. The same approach is relevant to evaporative pattern casting when the pattern material exhibits temperature-dependent relaxation.
The full numerical route can be summarized as follows. First, I used a mold-filling simulation to calculate the melt front, pressure history, temperature field, and packing state. Second, I imported the resulting temperature and stress state into a finite element solver. Third, I calculated in-mold cooling under mold constraint. Fourth, I released the mold constraint to simulate demolding. Fifth, I calculated post-demolding cooling and long-term viscoelastic deformation. This sequence is important because the mold constraint suppresses free shrinkage, and the subsequent release produces a sudden redistribution of stress and strain. In evaporative pattern casting, an analogous release can occur when a pattern is removed from a die or when a foam pattern is aged before coating.
To make the material description transparent, I list the main measured and fitted properties in a structured form. I used the transition temperature to define the solidification threshold. I used thermal conductivity and specific heat to calculate transient heat transfer. I used the pressure-volume-temperature relation to calculate shrinkage, compressibility, and thermal expansion. I used dynamic shear data to obtain the relaxation spectrum. I used tensile testing to estimate the Poisson ratio. These data formed the material parameter database for the simulation.
| Property | Value | Unit |
|---|---|---|
| Transition temperature | 49 | °C |
| Thermal conductivity, solid | 0.24 | W/(m·K) |
| Thermal conductivity, liquid | 0.14 | W/(m·K) |
| Specific heat, solid | 2.1 | kJ/(kg·K) |
| Specific heat, liquid | 3.2 | kJ/(kg·K) |
| Poisson ratio | 0.47 | — |
For the filling stage, I used the Cross-WLF viscosity model because wax viscosity depends strongly on temperature, shear rate, and pressure. The model is written as
$$
\eta=\frac{\eta_0}{1+\left(\frac{\eta_0\dot{\gamma}}{\tau^*}\right)^{1-n}}
$$
where \(\eta\) is the melt viscosity, \(\eta_0\) is the zero-shear viscosity, \(\dot{\gamma}\) is the shear rate, \(\tau^*\) is the critical stress for shear thinning, and \(n\) is the power-law index. The zero-shear viscosity is temperature and pressure dependent:
$$
\eta_0=D_1\exp\left[-\frac{A_1(T-T^*)}{A_2+(T-T^*)}\right]
$$
$$
T^*=D_2+D_3p,\quad A_2=\tilde{A}_2+D_3p
$$
The fitted Cross-WLF parameters I obtained are listed below. These values were used in the mold-filling simulation to calculate the pressure and temperature history of the wax.
| Parameter | Value | Unit |
|---|---|---|
| \(n\) | 0.3691 | — |
| \(\tau^*\) | 0.427887 | Pa |
| \(D_1\) | \(2.18\times10^{36}\) | Pa·s |
| \(D_2\) | 263.15 | K |
| \(D_3\) | 0 | K/Pa |
| \(A_1\) | 123.92 | — |
| \(A_2\) | 51.6 | K |
For the pressure-volume-temperature relation, I used a modified dual-domain Tait equation. This equation is important because it connects temperature, pressure, and specific volume, and therefore allows me to predict shrinkage and thermal expansion. The general form is
$$
V(T,P)=V_0(T)\left\{1-C\ln\left[1+\frac{P}{B(T)}\right]\right\}+V_1(T,P)
$$
where \(C\) is a universal constant, \(V_0\) is the specific volume at zero pressure, and \(B\) is the pressure sensitivity coefficient. Below the transition temperature, I used
$$
V_0=b_{1s}+b_{2s}(T-b_5),\quad B(T)=b_{3s}\exp[-b_{4s}(T-b_5)],\quad V_1=b_7\exp[b_8(T-b_5)-b_9P]
$$
Above the transition temperature, I used
$$
V_0=b_{1m}+b_{2m}(T-b_5),\quad B(T)=b_{3m}\exp[-b_{4m}(T-b_5)],\quad V_1=0
$$
The fitted Tait parameters are summarized in the following table. The coefficient of determination for both the solid and melt regions exceeded 0.99, so I considered the fit reliable for shrinkage calculation.
| Coefficient | Value | Unit |
|---|---|---|
| \(b_{1m}\) | 0.00114 | m³/kg |
| \(b_{2m}\) | \(1.372\times10^{-6}\) | m³/(kg·K) |
| \(b_{3m}\) | \(6.4005\times10^{7}\) | Pa |
| \(b_{4m}\) | 0.1685 | 1/K |
| \(b_{1s}\) | 0.001141 | m³/kg |
| \(b_{2s}\) | \(3.478\times10^{-6}\) | m³/(kg·K) |
| \(b_{3s}\) | \(3.9838\times10^{7}\) | Pa |
| \(b_{4s}\) | 0.0296 | 1/K |
| \(b_5\) | 324.15 | K |
| \(b_6\) | \(5\times10^{-7}\) | K/Pa |
| \(b_7\) | 0 | m³/kg |
| \(b_8\) | 0 | 1/K |
| \(b_9\) | 0 | 1/Pa |
The linear thermal expansion coefficient can be derived from the Tait equation. I used
$$
\kappa_T=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T
$$
$$
\alpha=\frac{1}{3V}\left(\frac{\partial V}{\partial T}\right)_P=\frac{1}{3}\left(\frac{b_2}{V_0}-P\kappa_Tb_4\right)
$$
For the viscoelastic stage, I adopted a generalized Maxwell model. The stress response is expressed as an integral over the strain history:
$$
\sigma_{ij}(t)=\int_{-\infty}^{t}G(\xi-\xi’)\frac{\partial e_{ij}(t)}{\partial t}dt+\delta_{ij}\int_{-\infty}^{t}K(\xi-\xi’)\frac{\partial(\varepsilon_m(t)-\varepsilon_{th})}{\partial t}dt
$$
where \(G\) is the shear relaxation modulus, \(K\) is the bulk relaxation modulus, \(e_{ij}\) is the deviatoric strain, \(\varepsilon_m\) is the mean strain, and \(\varepsilon_{th}\) is the thermal strain. The shear and bulk relaxation moduli are represented by Prony series:
$$
G(t)=G_\infty+\sum_{i=1}^{n}G_i\exp(-t/\tau_i)
$$
$$
K(t)=K_\infty+\sum_{i=1}^{n}K_i\exp(-t/\tau_i)
$$
To obtain the relaxation spectrum, I used the time-temperature superposition principle. The master curve relation is
$$
E(t,T)=E(t/a_T,T_0)
$$
where \(a_T\) is the shift factor and \(T_0\) is the reference temperature. For the temperatures used in my study, the Arrhenius form gave a good fit:
$$
\ln a_T=\frac{E_a}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)
$$
I fitted the dynamic storage and loss moduli with 15 Maxwell elements. The normalized relaxation spectrum is listed below. This spectrum was used in the finite element model to describe stress relaxation and creep during cooling and storage. In evaporative pattern casting, a similar spectrum could be used if the pattern material shows time-dependent deformation under thermal exposure.
| No. | \(g_i\) | \(k_i\) | \(\tau_i\) |
|---|---|---|---|
| 1 | 0.249872 | 0 | \(3.05\times10^{-6}\) |
| 2 | 0.108686 | 0 | \(4.17\times10^{-5}\) |
| 3 | 0.100223 | 0 | \(4.26\times10^{-5}\) |
| 4 | 0.075527 | 0 | 0.000302 |
| 5 | 0.048213 | 0 | 0.000635 |
| 6 | 0.043825 | 0 | 0.00132 |
| 7 | 0.070928 | 0 | 0.005115 |
| 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 | 2.25580 |
| 13 | 0.037882 | 0 | 25.5829 |
| 14 | 0.040110 | 0 | 10176.7 |
| 15 | 0.000997 | 0 | 277165 |
My numerical workflow separated the process into stages because each stage has different mechanical and thermal boundary conditions. During filling and packing, the wax behaves as a non-Newtonian fluid and the cavity pressure is controlled by the injection machine. During in-mold cooling, the wax has solidified but is still constrained by the mold. During demolding, the constraint is removed almost instantly and the stress field redistributes. During storage, the wax is free but still subject to gravity, contact with a support surface, and time-dependent relaxation. This staged approach is also useful for evaporative pattern casting, where pattern filling, cooling, ejection, and aging are distinct steps.
| Stage | Primary Solver | Key Physics | Boundary Condition |
|---|---|---|---|
| Filling and packing | Mold-filling solver | Non-Newtonian flow, heat transfer, compressibility | Injection pressure and mold wall |
| In-mold cooling | Finite element solver | Transient heat conduction, viscoelastic relaxation | Mold constraint |
| Demolding | Finite element solver | Stress release, strain redistribution | Constraint removal |
| Storage | Finite element solver | Cooling, creep, stress relaxation | Free surface and support contact |
For the demonstration geometry, I selected a three-way pipe wax pattern. I designed the mold, generated a three-dimensional mesh, and placed the gate near the center. The filling simulation showed a smooth melt front. Because the thermal conductivity of the wax is low, the center of the cavity remained hotter than the wall. This produced a fountain-like flow front, and the simulation resolved the characteristic spherical front. I also observed a race-track effect in the thicker sections: the thicker regions remained at higher temperature, had lower viscosity, and eventually filled faster than the thinner central regions. These flow features are important because they influence local temperature, local density, and local residual stress. In evaporative pattern casting, a similar interaction between flow front and thickness variation can affect pattern density and dimensional stability.
The process parameters I used for the baseline filling and packing simulation are listed below. I kept these values fixed for the initial validation and later varied them in the optimization study.
| Parameter | Value | Unit |
|---|---|---|
| Wax injection temperature | 55 | °C |
| Mold temperature | 22 | °C |
| Filling time | 5.5 | s |
| Packing pressure | 1.5 | MPa |
| Packing time | 40 | s |
| In-mold cooling time | 300 | s |
The filling simulation produced a total filling time of about 5.575 s. The melt-front temperature varied only slightly, which indicated that the filling process was thermally stable. The velocity-pressure switch occurred when the cavity was nearly full. After switchover, the packing pressure was held at 1.5 MPa. I used the pressure history at the gate as a boundary condition for the subsequent stress analysis. The gate pressure increased during filling because the flow resistance increased, and then it remained nearly constant during packing. When the gate froze, further packing had little effect, which is why packing time must be chosen carefully.
I validated the flow pattern with a partial-filling experiment. I stopped injection before the cavity was full and removed the partially filled wax after it had solidified. The experimental flow front agreed well with the simulated flow front. In both cases, the wax advanced from the gate toward the ends of the three-way pipe, and the thicker sections filled later but then caught up because of lower flow resistance. This validation gave me confidence that the filling and packing simulation provided a reasonable initial temperature and pressure field for the stress analysis.
For the stress analysis, I first calculated the temperature field through the entire cooling history. I then used that temperature history as input to the mechanical model. I treated the mold as a rigid body because the thermal expansion of the metal mold is much smaller than that of the wax. The contact between wax and mold was defined as hard contact in the normal direction. This simplification reduced convergence difficulty and allowed me to focus on the viscoelastic response of the wax. In evaporative pattern casting, a similar rigid-mold assumption can be used when the die is much stiffer than the pattern material.
During in-mold cooling, I compared an elastic model with a viscoelastic model. The viscoelastic model showed clear stress relaxation. The stress at representative points decreased by about 30% during in-mold cooling. The deformation was still constrained by the mold, so the displacement difference between the two models was not large before demolding. However, the stress state was very different. This difference became important at demolding, because the stored elastic energy and the residual stress distribution controlled the immediate warpage. In evaporative pattern casting, the same distinction between elastic and viscoelastic behavior can influence pattern distortion after ejection.
| Model | Maximum Deformation after Demolding | Unit | Stress Relaxation during Cooling |
|---|---|---|---|
| Elastic | 1.527 | mm | Not included |
| Viscoelastic | 1.279 | mm | About 30% |
| Difference | About 19% | — | — |
At demolding, I removed the mold constraint and let the wax deform freely. The stress field redistributed within a very short time. The displacement field also changed almost instantaneously. The three-way pipe tended to shrink toward the short branch. This happened because the junction between the long branch and the short branch was thicker, cooled more slowly, and therefore had a larger thermal contraction. The viscoelastic model produced less warpage than the elastic model because part of the residual stress had already relaxed before demolding. This result is important for tolerance control, and it also has implications for evaporative pattern casting, where residual stress relaxation can reduce or redistribute pattern distortion before coating.
After demolding, I placed the wax pattern horizontally and calculated 24 hours of storage. The pattern continued to cool to room temperature and then continued to deform by creep and stress relaxation. The deformation was not uniform. The side closer to the short branch deformed differently from the side farther away. I selected several key points on the long branch and the short branch to quantify the dimensional change. The measured and simulated dimensions agreed within 0.3 mm. This level of agreement supported the use of the full-process model for process optimization.
| Feature Size | Mold Dimension | Measured Dimension | Simulated Dimension | Deviation |
|---|---|---|---|---|
| Long-branch axial distance 1 | 170.5 mm | 168.62 mm | 168.50 mm | 0.12 mm |
| Long-branch axial distance 2 | 170.5 mm | 168.25 mm | 168.15 mm | 0.10 mm |
| Short-branch distance 1 | 117 mm | 115.53 mm | 115.29 mm | 0.24 mm |
| Short-branch distance 2 | 117 mm | 115.28 mm | 114.98 mm | 0.30 mm |
| Cross-branch distance 1 | 95 mm | 93.65 mm | 93.49 mm | 0.16 mm |
| Cross-branch distance 2 | 95 mm | 93.67 mm | 93.54 mm | 0.13 mm |
I also analyzed the contribution of each stage to the total deformation. The in-mold cooling stage contributed about 35% of the total dimensional change. The demolding stage contributed about 40%. The storage stage contributed about 25%. More than 98% of the total deformation occurred within the first 9 hours after demolding. This means that the period immediately after demolding is the most critical for dimensional control. If a wax pattern is measured too soon, it may still be changing. If it is handled too roughly, the residual stress distribution may be altered. In evaporative pattern casting, the same principle applies: the period immediately after pattern ejection can determine the final pattern tolerance.
| Stage | Contribution to Total Deformation | Unit |
|---|---|---|
| In-mold cooling | 35 | % |
| Demolding | 40 | % |
| Storage | 25 | % |
| Time to reach 98% deformation | 9 | h |
After validating the full-process model, I investigated how injection process parameters affect dimensional deformation. I selected five parameters: wax injection temperature, mold temperature, filling time, packing pressure, and packing time. I first used the Taguchi method to screen the parameters. The quality metrics were volumetric shrinkage and maximum deformation. I treated both as smaller-is-better characteristics. The signal-to-noise ratio was calculated as
$$
S/N=-10\log\left(\frac{1}{n}\sum_{i=1}^{n}y_i^2\right)
$$
The Taguchi design used two levels for each factor. The factor levels are listed below. This design allowed me to rank the influence of each parameter with a small number of simulations. I also note that a similar screening strategy can be used in evaporative pattern casting when many process variables affect pattern dimensions.
| Factor | Level 1 | Level 2 | Unit |
|---|---|---|---|
| Mold temperature | 19 | 25 | °C |
| Packing pressure | 1.5 | 3.5 | MPa |
| Filling time | 4 | 7 | s |
| Wax injection temperature | 55 | 65 | °C |
| Packing time | 20 | 40 | s |
The Taguchi results showed that filling time had the smallest effect on both volumetric shrinkage and deformation. For volumetric shrinkage, the influence order was mold temperature, wax injection temperature, packing time, packing pressure, and filling time. For deformation, the influence order was wax injection temperature, packing pressure, mold temperature, packing time, and filling time. This ranking told me that temperature and pressure control were more important than filling speed for the final dimensional stability. In evaporative pattern casting, a similar conclusion often holds: thermal history and pressure history dominate pattern shrinkage.
| Run | Mold Temperature | Packing Pressure | Filling Time | Wax Temperature | Packing Time | 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 mean response and signal-to-noise response confirmed the ranking. For shrinkage, the deltas were largest for mold temperature and wax injection temperature. For deformation, the deltas were largest for wax injection temperature and packing pressure. Because filling time had the smallest effect, I removed it from the subsequent response surface study. I fixed the filling time at 6 s, which was within the stable filling window. This decision reduced the number of design variables and made the optimization more efficient. In evaporative pattern casting, a similar variable screening step can reduce experimental cost and improve process understanding.
| Metric | Mold Temperature | Packing Pressure | Filling Time | Wax Temperature | Packing Time |
|---|---|---|---|---|---|
| Shrinkage mean, level 1 | 6.666 | 7.257 | 7.225 | 6.703 | 7.300 |
| Shrinkage mean, level 2 | 7.623 | 7.032 | 7.064 | 7.586 | 6.989 |
| Shrinkage delta | 0.957 | 0.225 | 0.161 | 0.882 | 0.312 |
| Shrinkage rank | 1 | 4 | 5 | 2 | 3 |
| Deformation mean, level 1 | 1.287 | 1.314 | 1.305 | 1.275 | 1.311 |
| Deformation mean, level 2 | 1.313 | 1.286 | 1.295 | 1.325 | 1.289 |
| Deformation delta | 0.027 | 0.028 | 0.010 | 0.051 | 0.022 |
| Deformation rank | 3 | 2 | 5 | 1 | 4 |
After screening, I used a central composite surface design with four factors: mold temperature, packing pressure, wax injection temperature, and packing time. I fitted a second-order response surface model. The general form was
$$
y=a_0+\sum_{i=1}^{n}a_ix_i+\sum_{i<j}a_{ij}x_ix_j+\sum_{i=1}^{n}a_{ii}x_i^2+\varepsilon $$=""
The factor levels for the response surface design are shown below. I used numerical simulation instead of physical experiments for the design points, so the center point could be evaluated once. This design allowed me to estimate linear effects, quadratic effects, and two-factor interactions. In evaporative pattern casting, response surface methods can likewise be used to optimize pattern quality when the pattern material is sensitive to thermal and pressure history.
| Factor | Low Level (-1) | Center (0) | High Level (+1) | Unit |
|---|---|---|---|---|
| Mold temperature | 19 | 22 | 25 | °C |
| Packing pressure | 1.5 | 2.5 | 3.5 | MPa |
| Wax injection temperature | 55 | 60 | 65 | °C |
| Packing time | 20 | 40 | 60 | s |
The design matrix and the simulated responses are given below. The volumetric shrinkage and deformation were recorded for each run. The results were used to fit the regression equations and to analyze the effects of each factor and interaction.
| Run | Mold Temperature | Packing Pressure | Wax Temperature | Packing Time | 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 analysis of variance for volumetric shrinkage showed that wax injection temperature, mold temperature, packing time, the interaction between wax injection temperature and packing time, packing pressure, and the quadratic term of packing time were significant. The fitted reduced model was
$$
\phi=-3.314+0.11631A-0.1048B+0.1483C+0.0554D+0.000326D^2-0.001499CD
$$
where \(A\) is mold temperature, \(B\) is packing pressure, \(C\) is wax injection temperature, and \(D\) is packing time. The adjusted coefficient of determination was 0.9504, and the predicted coefficient of determination was 0.9286. This indicated that the model had good predictive ability. The significant interaction between wax injection temperature and packing time showed that the benefit of longer packing time depended on the gate freeze time. When the wax temperature was higher, the gate froze later, so packing pressure could act for a longer time. When the wax temperature was lower, the gate froze earlier, and additional packing time had little effect. This behavior is similar to what can occur in evaporative pattern casting when the pattern material solidifies or hardens at different rates.
| Source | Degrees of Freedom | Adjusted Sum of Squares | Adjusted Mean Square | F Value | P Value |
|---|---|---|---|---|---|
| Model | 6 | 6.86218 | 1.1437 | 77.70 | 0.000 |
| A | 1 | 2.19144 | 2.19144 | 148.89 | 0.000 |
| B | 1 | 0.19778 | 0.19778 | 13.44 | 0.002 |
| C | 1 | 3.51461 | 3.51461 | 238.78 | 0.000 |
| D | 1 | 0.51302 | 0.51302 | 34.85 | 0.000 |
| \(D^2\) | 1 | 0.08584 | 0.08584 | 5.83 | 0.027 |
| \(CD\) | 1 | 0.35949 | 0.35949 | 24.42 | 0.000 |
| Error | 18 | 0.26494 | 0.01472 | — | — |
| Total | 24 | 7.12712 | — | — | — |
The analysis of variance for deformation showed that wax injection temperature, packing pressure, packing time, mold temperature, the interaction between packing pressure and wax injection temperature, and the interaction between wax injection temperature and packing time were significant. The reduced regression equation was
$$
\delta=1.0185+0.001394A-0.0883B+0.00505C+0.00484D+0.001207BC-0.000088CD
$$
The adjusted coefficient of determination was 0.9186, and the predicted coefficient of determination was 0.8773. The model was therefore slightly less accurate for deformation than for shrinkage, but still useful for process optimization. The significant interaction \(BC\) showed that packing pressure and wax injection temperature jointly affected warpage. A higher packing pressure reduced deformation, but the benefit depended on the thermal state of the melt. A higher wax injection temperature increased deformation, especially when packing pressure was low. In evaporative pattern casting, a similar interaction can appear between pattern temperature and compaction pressure.
| Source | Degrees of Freedom | Adjusted Sum of Squares | Adjusted Mean Square | F Value | P Value |
|---|---|---|---|---|---|
| Model | 6 | 0.017414 | 0.002902 | 46.15 | 0.000 |
| A | 1 | 0.000315 | 0.000315 | 5.01 | 0.038 |
| B | 1 | 0.004534 | 0.004534 | 72.08 | 0.000 |
| C | 1 | 0.009329 | 0.009329 | 148.33 | 0.000 |
| D | 1 | 0.001415 | 0.001415 | 22.49 | 0.000 |
| \(BC\) | 1 | 0.000583 | 0.000583 | 9.27 | 0.007 |
| \(CD\) | 1 | 0.001239 | 0.001239 | 19.70 | 0.000 |
| Error | 18 | 0.001132 | 0.000063 | — | — |
| Total | 24 | 0.018546 | — | — | — |
The response surface results showed clear monotonic trends for some factors and non-monotonic trends for others. Lower mold temperature reduced shrinkage. Lower wax injection temperature reduced both shrinkage and deformation. Higher packing pressure reduced deformation. Longer packing time reduced shrinkage up to a point, but after the gate froze, additional packing time had little effect. This is a critical insight for cycle time reduction. The same idea applies to evaporative pattern casting: after the pattern surface has sealed or hardened, additional holding pressure may not penetrate the pattern interior.
For multi-objective optimization, I used the reduced response surface models as objective functions. The optimization problem was
$$
\min F(x)=[f_1(x),f_2(x),\ldots,f_k(x)]
$$
$$
\mathrm{subject\ to}\quad g_i(x)\le 0,\quad h_i(x)=0,\quad a_i\le x_i\le b_i
$$
I used a non-dominated sorting genetic algorithm to find the Pareto front. The design variables were mold temperature, packing pressure, wax injection temperature, and packing time. The objective functions were volumetric shrinkage and deformation. The constraints were the allowable process windows. The Pareto front showed that shrinkage and deformation were competing objectives. In general, reducing shrinkage by lowering temperature could increase the risk of incomplete filling, while increasing packing pressure could reduce deformation but might increase residual stress. The Pareto set therefore provided a family of acceptable compromises rather than a single optimum.
From the Pareto front, I selected the point with the lowest practical shrinkage while maintaining deformation close to its minimum. The corresponding parameters were a mold 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. I then ran a validation simulation with these parameters. The predicted shrinkage was 6.1248%, and the predicted deformation was 1.246 mm. The simulation gave a shrinkage of 6.1661% and a deformation of 1.264 mm. The relative errors were 1.33% and 1.42%, respectively. This agreement confirmed that the combination of response surface modeling and genetic algorithm optimization was reliable for this wax injection molding problem. The same optimization architecture can be transferred to evaporative pattern casting when pattern shrinkage and distortion are the main quality metrics.
| Metric | Numerical Simulation | Genetic Algorithm Prediction | Relative Error |
|---|---|---|---|
| Volumetric shrinkage | 6.1661% | 6.1248% | 1.33% |
| Deformation | 1.264 mm | 1.246 mm | 1.42% |
| Mold temperature | 19 °C | 19 °C | — |
| Packing pressure | 3.5 MPa | 3.5 MPa | — |
| Wax injection temperature | 55 °C | 55 °C | — |
| Packing time | 41.5 s | 41.5 s | — |
In my view, the most important result is that the full-process viscoelastic simulation changes the interpretation of wax pattern quality. A purely elastic analysis can overestimate warpage because it ignores stress relaxation. A purely filling analysis can miss the long-term storage deformation because it stops at the end of packing. A purely experimental trial-and-error approach can be slow and expensive because it cannot easily separate the contributions of in-mold cooling, demolding, and storage. By combining filling simulation, heat transfer analysis, viscoelastic stress analysis, and statistical optimization, I obtained a more complete picture. I also believe that this framework has value beyond investment casting. In evaporative pattern casting, the pattern is also a sacrificial, dimensionally critical component. The pattern experiences filling or forming, cooling, ejection, and storage before it is coated or embedded. The same stages, the same relaxation mechanisms, and the same optimization logic can be applied. For evaporative pattern casting, the material model would need to represent foam or polymer behavior, but the computational workflow is essentially the same.
I also observed that the junction regions of the three-way pipe were the most sensitive to residual stress. These regions were thicker, cooled more slowly, and had a larger thermal gradient. When the mold constraint was removed, these regions deformed first and most strongly. This suggests that local geometry modification, local cooling control, or local support fixtures could be used to reduce warpage. In evaporative pattern casting, local geometry and local density variation can produce similar hot spots or weak spots, so local control strategies are also relevant. The use of fixtures, supports, or controlled cooling after ejection can help stabilize the pattern before subsequent operations.
Another important point is that filling time had the smallest effect on final shrinkage and deformation. This does not mean that filling time is unimportant. If filling time is too short, the flow can become unstable and can trap air. If filling time is too long, the melt can freeze before the cavity is full. However, within the stable processing window, filling time is less influential than temperature and pressure. I therefore fixed filling time at 6 s and focused the optimization on thermal and pressure parameters. This is a practical result because it simplifies process control. For evaporative pattern casting, a similar stable filling window should be identified first, and then the optimization can focus on the most influential variables.
The Taguchi method and response surface method worked well together. The Taguchi method efficiently screened out the least important factor. The response surface method then provided a continuous mathematical model of the remaining factors. The genetic algorithm then searched the model for Pareto-optimal solutions. This three-step strategy is efficient and robust. It reduces the number of physical trials or simulations, it quantifies interactions, and it provides a set of trade-off solutions rather than a single point. I would recommend this strategy for any pattern-making process with multiple quality metrics, including evaporative pattern casting.
In terms of material modeling, the generalized Maxwell model captured the stress relaxation that occurred during in-mold cooling and storage. The relaxation spectrum covered a wide range of relaxation times, from about \(3\times10^{-6}\) s to about \(2.8\times10^5\) s. This broad spectrum was necessary because the wax relaxes quickly at high temperature and slowly at low temperature. The time-temperature superposition principle allowed me to construct the master curve from dynamic tests at several temperatures. Without this spectrum, the simulation would have overpredicted residual stress and warpage. In evaporative pattern casting, if the pattern material is polymeric, a similar broad spectrum may be needed.
The pressure-volume-temperature relation was also essential. The modified dual-domain Tait equation described the specific volume as a function of temperature and pressure. This allowed me to calculate shrinkage from the thermodynamic state rather than from a single constant shrinkage value. The Tait equation also provided the pressure sensitivity and thermal expansion coefficient. These quantities are important because wax shrinkage depends on both cooling and packing. In evaporative pattern casting, the pressure-volume-temperature behavior of the foam or polymer controls pattern density and dimensional change. Therefore, measuring and fitting the pressure-volume-temperature relation should be a priority in any pattern material study.
The gate freeze time was a key hidden variable. The response surface results showed that packing time interacted strongly with wax injection temperature. When the wax temperature was higher, the gate remained open longer, so packing pressure could compensate for shrinkage for a longer period. When the wax temperature was lower, the gate froze earlier, and additional packing time became ineffective. This explains why the quadratic term for packing time was significant in the shrinkage model. It also explains why the optimal packing time was about 41.5 s rather than the maximum value. In evaporative pattern casting, the gate or inlet freeze time can similarly control how long pressure can be transmitted into the pattern.
My validation experiments showed that the simulated dimensions were slightly smaller than the measured dimensions. The deviation was less than 0.3 mm. This small systematic difference could come from several sources: measurement uncertainty, temperature control error, mold compliance, or slight differences between the fitted material parameters and the actual material. Even with these uncertainties, the model was accurate enough to rank process parameters and to identify the optimal window. The fact that the model captured the trend and the magnitude of deformation is more important than matching every point exactly. For industrial application, the model can be calibrated with a small number of measured dimensions and then used for optimization.
I also note that the storage stage is often neglected in practice. Operators may measure a wax pattern immediately after demolding and assume that the dimension is final. My results show that about 25% of the total deformation occurs after demolding. If the pattern is measured too early, the measurement will not represent the stabilized dimension. If the pattern is assembled into a tree too early, the residual stress state may be different from the stabilized state. In evaporative pattern casting, the same risk exists: a pattern measured or assembled too soon may continue to change dimension during coating, handling, or pouring. Therefore, a controlled aging period is beneficial. The aging time should be based on the relaxation spectrum and the storage temperature.
Based on my results, I recommend the following practical guidelines for wax injection molding. First, measure the pressure-volume-temperature relation and the dynamic modulus of the wax before designing the mold. Second, use a full-process simulation rather than a single-stage simulation. Third, screen process parameters with a Taguchi design to identify the most influential factors. Fourth, use a response surface design to model interactions. Fifth, use a multi-objective genetic algorithm to find a Pareto front. Sixth, validate the optimized parameters with a small number of physical trials. These guidelines are also applicable to evaporative pattern casting, although the material model and the thermal boundary conditions would need to be adapted.
For future work, I plan to extend the model to larger and more complex geometries. The three-way pipe was useful because it contained thickness variation, a junction, and two branches, but it did not contain thin walls, long flow paths, or inserts. A complex thin-walled component would test the model more severely. I also plan to include mold compliance and insert restraint in the model. In investment casting, ceramic cores and metal inserts can constrain wax shrinkage, and these constraints can change the residual stress field. In evaporative pattern casting, embedded inserts or coatings can play a similar role. Including these effects would improve the predictive capability.
I also plan to develop a dimensional compensation strategy. The current model predicts deformation for a given mold shape. The next step is to invert the model: given a target wax pattern shape, calculate the mold shape that compensates for shrinkage and warpage. This is a reverse design problem. It can be solved by coupling the finite element model with an optimization algorithm. The compensation would be applied to the mold cavity. In evaporative pattern casting, a similar reverse design could be used to compensate for pattern shrinkage and distortion. This would reduce trial-and-error and shorten the development cycle.
Finally, I want to emphasize the role of evaporative pattern casting in the broader context of sacrificial pattern processes. Investment casting and evaporative pattern casting both rely on a pattern that is destroyed or removed after mold formation. In both processes, the pattern must be dimensionally stable during handling and storage. In both processes, the pattern material can exhibit viscoelastic behavior. In both processes, the final casting tolerance depends on the accumulated error from pattern formation, pattern cooling, pattern removal, mold formation, and metal solidification. My work shows that a staged, viscoelastic, full-process simulation can provide quantitative insight into these errors. Although my experimental material was an investment casting wax, the computational framework is general. With appropriate material parameters, it can be applied to evaporative pattern casting and other pattern-based casting processes.
In summary, I constructed a full-process numerical model for wax injection molding in investment casting. The model included Cross-WLF viscosity, modified dual-domain Tait pressure-volume-temperature behavior, generalized Maxwell viscoelasticity, time-temperature superposition, and staged finite element analysis. I validated the flow pattern with partial-filling experiments and validated the final dimensions with measured wax patterns. I found that in-mold cooling, demolding, and storage contributed about 35%, 40%, and 25% of the total deformation, respectively. More than 98% of the deformation occurred within 9 hours after demolding. I screened process parameters with the Taguchi method, modeled the response with a central composite design, and optimized the process with a non-dominated sorting genetic algorithm. The optimal parameters were a mold 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. The validation simulation agreed with the optimization prediction within about 1.4%. I believe this approach can support the design and optimization of wax injection molding and can also provide a useful reference for evaporative pattern casting and related sacrificial pattern processes.