Investment casting is used for aerospace, automotive, medical, and power-generation components because it can produce complex, thin-walled shapes with good surface finish and dimensional control. The process is not a single forming operation but a chain of dimensional transfers. The wax pattern defines the cavity in the ceramic shell; the shell defines the mold cavity for the alloy; solidification and cooling then add further contraction. Dimensional errors introduced at any step are inherited by the next, so the wax pattern is the first major tolerance source in the chain.
Wax injection includes filling and packing, in-mold cooling, demolding, and post-demold storage. The first three stages occur under mold constraint; the last occurs with little or no constraint. Because wax is viscoelastic, stress relaxation and creep continue after ejection. A pattern that measures correctly at demolding can still drift out of tolerance during storage. Full-process simulation therefore has to connect flow-induced stress, thermal stress, demolding release, and time-dependent storage deformation rather than compute only final free shrinkage.
The Dimensional Error Chain Begins with the Wax Pattern
In investment casting, the wax pattern is not a disposable intermediate. Its dimensions determine the shell cavity, and the shell cavity determines the final casting. Studies of the process chain have identified wax-pattern shrinkage and warpage, together with alloy solidification deformation, as major contributors to final dimensional error. The wax-pattern contribution can be divided into four stages:
- Filling and packing: non-Newtonian flow, fountain flow, gate freeze, and pressure history establish the initial stress state.
- In-mold cooling: constrained shrinkage under a thermal gradient produces residual stress, part of which relaxes before ejection.
- Demolding: loss of mold constraint causes rapid stress redistribution and warpage.
- Post-demold storage: cooling to ambient, creep, and relaxation continue to change dimensions after the pattern has been removed.
Die manufacturing error is systematic and can be reduced by machining accuracy. In-mold error is controlled mainly by injection parameters. Demold and storage errors are governed by viscoelasticity and are therefore more difficult to capture with a single shrinkage coefficient. Conventional mold-flow analysis can predict filling and packing and can estimate free shrinkage from pressure–volume–temperature data and thermal expansion. For three-dimensional models, however, it generally assumes free warpage after ejection and does not resolve long-term relaxation under contact or fixturing. The corrected residual in-mold stress approach is also mainly associated with shell or dual-domain representations rather than a full transient viscoelastic stress history. This is the gap that a sequential mold-flow and finite-element workflow is intended to close.
Material Behavior: Viscosity, PVT, and Viscoelastic Relaxation
The simulations described here use K512 wax, a low-ash pattern wax with a linear shrinkage of 0.9–1.1% and good flowability. Its transition temperature is 49 °C, so the injection window is only a few degrees above solidification. Three material descriptions are required: a shear-thinning viscosity model for filling, a pressure–volume–temperature model for compressibility and thermal expansion, and a viscoelastic model for stress relaxation and creep.
For filling and packing, the Cross-WLF model is used because wax viscosity depends on temperature, pressure, and shear rate:
$$\eta=\frac{\eta_0}{1+\left(\eta_0\dot{\gamma}/\tau^{*}\right)^{1-n}},\qquad \eta_0=D_1\exp\left[-\frac{A_1(T-T^{*})}{A_2+(T-T^{*})}\right]$$
Here, \(n\) is the power-law exponent, \(\tau^{*}\) is the critical stress for shear thinning, and \(T^{*}=D_2+D_3p\). The parameter \(A_2\) also carries a pressure term. The fitted K512 parameters reproduce the shear-thinning branch and the temperature shift of the zero-shear viscosity.
The pressure–volume–temperature behavior is represented by a modified dual-domain Tait equation:
$$V(T,P)=V_0(T)\left\{1-C\ln\left[1+\frac{P}{B(T)}\right]\right\}+V_1(T,P)$$
where \(C=0.0894\) and separate coefficient sets are used below and above the transition line \(b_5+b_6P\). For K512, \(b_5=324.15\ \mathrm{K}\) and \(b_6=5\times10^{-7}\ \mathrm{K/Pa}\). The fit gives coefficients of determination above 0.99 and provides the specific volume, compressibility, and thermal expansion used in the stress calculation.
For the solid-like and storage stages, a generalized Maxwell model is used:
$$G(t)=G_\infty+\sum_{i=1}^{n}G_i e^{-t/\tau_i}$$
The relaxation spectrum was obtained from dynamic moduli measured at 30, 35, 45, 50, and 55 °C, shifted to a 45 °C reference using the Arrhenius form of the time–temperature superposition principle. A 15-element spectrum gives a fit with \(R^2=0.999\). The linear viscoelastic limit was set at 0.1% strain, and the Poisson ratio was measured as 0.47. The bulk response is treated as time-independent, so relaxation is carried mainly by the shear modulus.
| Parameter | Value | Unit |
|---|---|---|
| Transition temperature | 49 | °C |
| Poisson ratio | 0.47 | – |
| Cross-WLF exponent \(n\) | 0.3691 | – |
| Critical stress \(\tau^{*}\) | 0.427887 | Pa |
| Zero-shear coefficient \(D_1\) | 2.18 × 1036 | Pa·s |
| \(D_2\) | 263.15 | K |
| \(A_1\) | 123.92 | – |
| \(A_2\) | 51.6 | K |
| Tait transition \(b_5\) | 324.15 | K |
| Tait pressure coefficient \(b_6\) | 5 × 10-7 | K/Pa |
| Maxwell elements | 15 | – |
A Sequential MoldFlow–Abaqus Workflow for the Full Process
The simulation separates the problem into a mold-flow stage and a solid-mechanics stage. The coupling variables are the temperature field and the stress state at the end of packing. The workflow is sequential because a fully coupled thermal–mechanical simulation of the entire process is difficult to converge and unnecessarily expensive for this application.
- Filling and packing in MoldFlow. A three-dimensional mesh is used with the Cross-WLF viscosity model and the Tait PVT model. The analysis gives the flow front, pressure history, temperature field, and residual stress at the end of packing. The melt is assumed to remain in contact with the cavity under packing pressure.
- In-mold cooling in Abaqus. The temperature history is computed first, followed by the stress and displacement fields. The mold is treated as a rigid body, contact in the normal direction is hard, and mold thermal expansion is neglected. The cooling time is 300 s, and the wax is viscoelastic.
- Demolding. All mold constraints are removed instantaneously over 1 s. The demolding step uses an elastic description because the release is short compared with the relaxation time. Ejector-pin mechanics are not included.
- Post-demold storage. The pattern is placed on a horizontal plane and held for 24 h. The viscoelastic model continues to cool the pattern to room temperature and then captures relaxation and creep under gravity and contact.
Temperature–stress sequential coupling avoids the convergence problems of direct thermal–mechanical coupling. The mold constraint is represented by contact, and the storage stage is represented by contact with a rigid plane. This is a reasonable first-order model for a pattern that is not clamped or supported by a complex fixture.
Flow and Stress Evolution in a Three-Way Pipe Pattern
The demonstration part is a three-way pipe pattern with a center gate. The injection conditions are 55 °C melt temperature, 22 °C mold temperature, 5.5 s filling time, 1.5 MPa packing pressure, and 40 s packing time. In-mold cooling lasts 300 s, demolding is modeled over 1 s, and storage lasts 24 h.
The predicted flow front temperature ranges from 54.35 to 55.00 °C, a variation of only 0.65 °C. The maximum pressure at the velocity-to-pressure switch is 0.0545 MPa. The flow pattern shows the expected fountain effect: wax near the wall cools and stiffens, while the hotter central layer continues to move and turns outward at the front. The thicker section of the long branch also shows race tracking, because its higher average temperature and lower viscosity allow faster flow than in the thinner central region. Off-center short-shot experiments at 2, 3, and 4 s produced flow shapes consistent with the simulated fronts at 2.089, 3.081, and 3.963 s, supporting the flow model.
During in-mold cooling, stress at the gate point O1 and at an inner-wall point O2 at the same axial location falls by about 30%. This is direct evidence of stress relaxation under constraint. At demolding, the residual stress field redistributes within approximately \(10^{-4}\) s. The elastic model predicts a maximum deformation of 1.527 mm, whereas the viscoelastic model predicts 1.279 mm. The elastic assumption therefore overpredicts warpage by about 19% in this case. The difference arises because a substantial part of the thermal stress has already relaxed before the mold is opened.
After 24 h on a horizontal plane, the pattern deforms toward the short branch. The thick intersection between the long and short branches cools more slowly, reaches a higher average temperature, and would contract more if it were free. The mold constraint prevents that contraction during cooling; demolding releases the stored stress; storage adds time-dependent creep. For the key dimensional changes of this pattern, approximately 35% occurs during in-mold cooling, 40% during demolding, and 25% during storage. More than 98% of the total change occurs within the first 9 h after demolding.
| Dimension | XAE1 | XAE2 | YA12 | ZB12 | XC12 | ZD12 |
|---|---|---|---|---|---|---|
| Mold dimension | 170.5 | 170.5 | 117 | 117 | 95 | 95 |
| Measured | 168.62 | 168.25 | 115.53 | 115.28 | 93.65 | 93.67 |
| Simulated | 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 |
The overall contraction is about 2 mm, and the simulation remains within 0.3 mm of the measured dimensions. The gap between two top points on the long branch shows a non-monotonic history: it becomes negative and then more negative under mold constraint, jumps rapidly at demolding, and grows gradually during storage. That sequence is a signature of viscoelastic stress history. A single-stage shrinkage model cannot reproduce the constrained contraction, the release jump, and the subsequent creep in one consistent calculation.
Process-Window Screening and Multi-Objective Optimization
The process window considered here is 19–25 °C mold temperature, 1.5–3.5 MPa packing pressure, 4–7 s filling time, 55–65 °C injection temperature, and 20–60 s packing time. A two-level, five-factor Taguchi screen with eight runs was used first to rank the main effects on volume shrinkage and deformation.
| Factor | Δ shrinkage (percentage points) | Rank for shrinkage | Δ deformation (mm) | Rank for deformation |
|---|---|---|---|---|
| Mold temperature A | 0.957 | 1 | 0.027 | 3 |
| Packing pressure B | 0.225 | 4 | 0.028 | 2 |
| Filling time C | 0.161 | 5 | 0.010 | 5 |
| Injection temperature D | 0.882 | 2 | 0.051 | 1 |
| Packing time E | 0.312 | 3 | 0.022 | 4 |
Mold temperature and injection temperature dominate shrinkage, while injection temperature and packing pressure dominate deformation. Filling time is the least significant factor for both responses. A separate 2–9 s filling-time sweep showed that sink-mark depth reaches a minimum in the 4–7 s range, shrinkage varies non-monotonically, and clamping force generally rises as filling slows. A filling time of 6 s was therefore fixed for the subsequent optimization.
The four remaining factors were analyzed with a central composite face-centered design of 25 runs. Reduced quadratic response surfaces retained only statistically significant terms. The shrinkage model is dominated by the linear effects of mold temperature, packing pressure, injection temperature, and packing time, with a significant packing-time squared term and a significant injection-temperature × packing-time interaction. The deformation model is dominated by injection temperature, packing pressure, packing time, and mold temperature, with significant packing-pressure × injection-temperature and injection-temperature × packing-time interactions.
| Response | Adjusted \(R^2\) | Predicted \(R^2\) | Significant terms |
|---|---|---|---|
| Volume shrinkage \(\varphi\) | 0.9504 | 0.9286 | A, B, C, D, D², C×D |
| Deformation \(\delta\) | 0.9186 | 0.8773 | A, B, C, D, B×C, C×D |
The interactions are mechanistically consistent. For shrinkage, packing time has a saturation point because gate freeze cuts off pressure transmission. A higher injection temperature delays freeze, so packing pressure remains effective for a longer part of the cycle. For deformation, packing pressure and injection temperature jointly set the residual stress state, while packing time interacts with the thermal history. Lower mold and injection temperatures reduce shrinkage and deformation but raise viscosity and increase short-shot risk. Higher packing pressure reduces both responses but can cause flash and demolding difficulty. Longer packing time helps only until the gate freezes.
The two response surfaces were used as objective functions in an NSGA-II multi-objective genetic algorithm. The aim was to minimize both volume shrinkage and deformation within the tested process window. The Pareto front was nearly flat in deformation, so a representative point on the low-shrinkage side was selected. The corresponding parameters are 19 °C mold temperature, 3.5 MPa packing pressure, 55 °C injection temperature, 41.5 s packing time, and 6 s filling time.
| Quantity | Value |
|---|---|
| Mold temperature | 19 °C |
| Packing pressure | 3.5 MPa |
| Injection temperature | 55 °C |
| Packing time | 41.5 s |
| Filling time | 6 s |
| Predicted shrinkage \(\varphi\) | 6.1248% |
| Simulated shrinkage \(\varphi\) | 6.1661% |
| Shrinkage error | 1.33% |
| Predicted deformation \(\delta\) | 1.246 mm |
| Simulated deformation \(\delta\) | 1.264 mm |
| Deformation error | 1.42% |
Design Implications and Remaining Limits
The full-process model leads to three practical conclusions. First, demolding should not be treated as an elastic springback only. In the three-way pipe case, an elastic demolding calculation overpredicts warpage by about 19% because it ignores relaxation before ejection. Second, storage is not a minor correction. About one quarter of the dimensional change occurs after demolding, and most of it occurs within the first 9 h. Patterns that are measured immediately after ejection can therefore appear acceptable and still fail later. Third, process optimization should not rely on a single shrinkage index. Shrinkage and deformation have different sensitivities: mold and injection temperatures dominate shrinkage, while injection temperature and packing pressure dominate deformation. A multi-objective search is more appropriate than minimizing one response alone.
The analysis also has clear limits. The demonstration part is a three-way pipe, the mold is treated as rigid, ejector-pin mechanics are omitted, and storage is represented by a simple horizontal support. Complex thin-walled patterns may contact ceramic cores, fixtures, or supports during cooling, and those constraints can change the relaxation path. The response surfaces are local to the tested window and should not be extrapolated without verification. The next step is to combine the viscoelastic process model with fixture constraints and inverse deformation design, so that the mold contour can be compensated for the predicted shrinkage, warpage, and storage drift rather than only evaluated after the fact.