In modern manufacturing, pressure casting is a critical process for producing high-precision components, especially for lightweight alloys like magnesium. However, metal casting defects such as shrinkage porosity remain a significant challenge, affecting mechanical properties and product reliability. This study focuses on predicting these metal casting defects by integrating Smoothed Particle Hydrodynamics (SPH) and Finite Element Method (FEM) for numerical simulation. The SPH method handles the flow, temperature, and solidification fields during casting, while FEM computes the stress field under pressure. By incorporating the Niyama criterion into the SPH framework, we accurately predict shrinkage defects, validated against commercial software like ProCAST. The coupling approach accounts for pressure-induced temperature variations, enhancing prediction accuracy for metal casting defects. This work demonstrates the robustness of SPH-FEM coupling in addressing complex metal casting defects in pressure casting applications.
The SPH method, a mesh-free Lagrangian approach, excels in modeling free-surface flows and large deformations, making it ideal for simulating metal casting processes. Its adaptability allows for precise tracking of fluid particles during filling, which is crucial for identifying potential metal casting defects. The core equations governing SPH include the conservation laws discretized using kernel functions. For instance, the density evolution is given by:
$$ \frac{d\rho_i}{dt} = \sum_{j=1}^{N} m_j \mathbf{v}_{ij} \cdot \nabla_i W_{ij} $$
where $\rho_i$ is the density of particle $i$, $m_j$ is the mass of neighboring particle $j$, $\mathbf{v}_{ij}$ is the velocity difference, and $W_{ij}$ is the kernel function. The momentum equation accounts for pressure and viscous forces:
$$ \frac{d\mathbf{v}_i}{dt} = -\sum_{j=1}^{N} m_j \left( \frac{P_i}{\rho_i^2} + \frac{P_j}{\rho_j^2} \right) \nabla_i W_{ij} + \sum_{j=1}^{N} m_j \frac{\mu_i + \mu_j}{\rho_i \rho_j} \mathbf{v}_{ij} \left( \frac{1}{r_{ij}} \frac{\partial W_{ij}}{\partial r_{ij}} \right) + \mathbf{F}^{(s)} $$
Here, $P$ represents pressure, $\mu$ is viscosity, and $\mathbf{F}^{(s)}$ includes gravity and surface tension. The energy equation models thermal effects:
$$ \frac{de_i}{dt} = \frac{1}{2} \sum_{j=1}^{N} m_j \left( \frac{P_i + P_j}{\rho_i \rho_j} \right) \mathbf{v}_{ij} \cdot \frac{\partial W_{ij}}{\partial r_{ij}} $$
These equations enable realistic simulation of molten metal behavior, directly influencing the formation of metal casting defects. For boundary conditions, virtual particles represent mold walls, ensuring accurate interactions. The kernel function used is the B-spline type, defined for 3D cases as:
$$ W(R, h) = \alpha_d \begin{cases}
\frac{2}{3} – R^2 + \frac{1}{2} R^3 & \text{for } 0 \leq R < 1 \\
\frac{1}{6} (2 – R)^3 & \text{for } 1 \leq R < 2 \\
0 & \text{for } R \geq 2
\end{cases} $$
with $\alpha_d = \frac{3}{2\pi h^3}$. This formulation ensures smooth particle interactions, critical for predicting metal casting defects like shrinkage porosity.
In parallel, FEM handles stress-strain analysis during solidification, providing insights into how pressure affects temperature and defect formation. The governing equation for FEM dynamics is:
$$ \mathbf{M} \ddot{\mathbf{u}}(t) + \mathbf{C} \dot{\mathbf{u}}(t) + \mathbf{K} \mathbf{u}(t) = \mathbf{Q}(t) $$
where $\mathbf{M}$ is the mass matrix, $\mathbf{C}$ is the damping matrix, $\mathbf{K}$ is the stiffness matrix, and $\mathbf{Q}$ is the load vector. The strain increments consider elastic, thermal, and inelastic components:
$$ \{\Delta \varepsilon\} = \{\Delta \varepsilon_{el}\} + \{\Delta \varepsilon_{th}\} + \{\Delta \varepsilon_{in}\} $$
For thermal strains, the change is computed as:
$$ \{\Delta \varepsilon_{th}^{t+\Delta t}\} = [\alpha_1(\theta^{t+\Delta t}) – \alpha_1(\theta^t)] \{1, 1, 0\}^T $$
with $\alpha_1(\theta) = \sqrt[3]{\frac{\rho(\theta_0)}{\rho(\theta)}} – 1$. The inelastic strain rate follows a viscoplastic model:
$$ \{\dot{\varepsilon}_{in}\} = \frac{3}{2} \bar{\varepsilon}_{in} \frac{\{\sigma’\}}{\bar{\sigma}} $$
where $\bar{\varepsilon}_{in}$ is the equivalent inelastic strain rate, and $\bar{\sigma}$ is the equivalent stress. For magnesium alloys, a hyperbolic sine relation describes the strain rate:
$$ \dot{\varepsilon} = A [\sinh(\alpha \sigma^*)]^n \exp\left(-\frac{Q}{R\theta}\right) $$
This coupled thermo-mechanical approach captures the essence of pressure casting, where stress influences temperature via:
$$ dT = \frac{T \Delta V}{\Delta H} dp $$
This equation shows how pressure changes affect temperature, refining the SPH-based prediction of metal casting defects. The integration of SPH and FEM involves mapping SPH particle data to FEM nodes using a radial search algorithm, ensuring seamless transition between filling and solidification phases.
To predict metal casting defects, specifically shrinkage porosity, the Niyama criterion is embedded in the SPH code. It evaluates the ratio of temperature gradient $G$ to cooling rate $R$:
$$ G = \sqrt{\left( \frac{\partial T}{\partial x} \right)^2 + \left( \frac{\partial T}{\partial y} \right)^2 + \left( \frac{\partial T}{\partial z} \right)^2} $$
$$ R = \frac{T_L – T_S}{t_L – t_S} $$
where $T_L$ and $T_S$ are liquidus and solidus temperatures, and $t_L$ and $t_S$ are corresponding times. The criterion states that shrinkage occurs if:
$$ N_y = \frac{G}{\sqrt{R}} < N_y^* $$
where $N_y^*$ is a critical value. This criterion effectively identifies regions prone to metal casting defects by highlighting areas with insufficient feeding during solidification.
For the simulation, a magnesium alloy AM60B bracket was modeled, with material properties and process parameters summarized in tables. The geometry includes a gating system to facilitate filling. Key parameters are listed below:
| Parameter | Value |
|---|---|
| Alloy | AM60B |
| Density Range | 1.78–1.82 g/cm³ |
| Liquidus Temperature | 596 °C |
| Solidus Temperature | 468 °C |
| Specific Heat | 0.696–0.916 kJ/kg·°C |
Process conditions for pressure casting are critical in minimizing metal casting defects:
| Parameter | Value |
|---|---|
| Plunger Pressure | 11 MPa |
| Plunger Velocity | 4 m/s |
| Pouring Temperature | 680 °C |
| Mold Preheat Temperature | 220 °C |
The SPH simulation used over 2 million particles, including virtual boundary particles, while FEM meshed the domain into 400,000 elements for stress analysis. Interface heat transfer coefficients were set to 2000 W/m²·K for metal-mold contact and 41.86 W/m²·K for mold-air interfaces, affecting temperature distribution and metal casting defects formation.
Results from the SPH simulation show the filling process completed in 0.0307 seconds, with molten metal splitting into multiple streams upon entering the cavity. The flow patterns indicate turbulent regions where metal casting defects might initiate. Temperature fields reveal rapid cooling in thin sections, while thicker areas retain heat longer, promoting shrinkage. The Niyama criterion applied in SPH identifies shrinkage porosity predominantly in central thick regions and near thin mold edges, where stress from FEM analysis is minimal. This aligns with ProCAST predictions, validating the SPH-FEM approach for metal casting defects assessment.
FEM stress results demonstrate that low-stress zones correlate with shrinkage locations, as insufficient pressure hinders molten metal feeding. The equation for stress-dependent temperature change underscores this interaction. For instance, in high-pressure zones, solidification is accelerated, reducing metal casting defects, whereas low-pressure areas exhibit higher defect propensity. The table below summarizes defect distribution based on Niyama values:
| Region | Niyama Value (℃⁰·⁵·s⁰·⁵/mm) | Defect Severity |
|---|---|---|
| Central Thick Area | < 0.5 | High |
| Thin Edge Zones | < 0.5 | Moderate |
| Other Regions | ≥ 0.5 | Low |
This comprehensive analysis confirms that SPH-FEM coupling effectively predicts metal casting defects by integrating multi-physics simulations. The method’s accuracy in capturing flow dynamics, thermal gradients, and stress responses makes it a powerful tool for optimizing casting processes and reducing defects in industrial applications.