AM Microstructure Simulation

It predicts crystal growth during solidification (columnar/equiaxed transition) using phase field methods or cellular automaton methods. The ratio of temperature gradient G to solidification rate R governs the microstructure.

Expressing this mathematically gives us this.

Phase field equation:

AM microstructure simulation is formulated as a coupled problem of thermodynamics, material mechanics, and fluid dynamics. Since the physical phenomena of manufacturing processes span multiple time and spatial scales, an appropriate combination of macro-scale continuum models and meso/micro-scale material models is required. The goal is to quantitatively predict the causal relationship between process parameters (temperature, speed, load, etc.) and product quality (dimensional accuracy, defects, mechanical properties).

The accuracy of manufacturing process simulation heavily depends on the fidelity of the material model. It is necessary to properly define elastoplastic constitutive laws, creep laws, phase transformation models, etc., as functions of temperature and strain rate. Data obtained from material testing (tensile, compression, torsion) is fitted, and validity within the extrapolation range is verified. Thermodynamic databases such as JMatPro and Thermo-Calc are also utilized.

Manufacturing process simulation is formulated as a coupled problem of thermodynamics, fluid dynamics, and solid mechanics.

Here, $T$ is temperature, $\mathbf{v}$ is the material velocity field, $k$ is thermal conductivity, and $Q$ is internal heat generation (Joule heating, latent heat, frictional heat, etc.).

During solidification, the release/absorption of latent heat significantly affects the temperature field. Formulation using the enthalpy method:

Expressing this mathematically gives us this.

Here, $L$ is latent heat, and $f_l(T)$ is the liquid fraction (takes a value between 0 and 1 in the solid-liquid coexistence region).

Plastic deformation of metals is described by constitutive laws such as the Johnson-Cook model:

$A$: Initial yield stress, $B$: Hardening coefficient, $n$: Hardening exponent, $C$: Strain rate sensitivity, $m$: Thermal softening exponent.

The flow of molten metal or resin follows the Navier-Stokes equations, but high viscosity and non-Newtonian fluid characteristics must be considered. The Cross-WLF model is standard for injection molding:

Here, $\eta_0$ is the zero-shear viscosity, $\tau^*$ is the critical shear stress, and $n$ is the power-law index.

Common assumptions include: continuum mechanics (ignoring atomic-scale effects), isotropic material properties, neglecting chemical reactions (except for specific processes), and assuming steady-state or quasi-steady conditions for certain process stages. Applicability limits arise when these assumptions break down, such as at very high strain rates, near material phase boundaries, or when microstructural evolution significantly affects macroscopic behavior.

Important ones include: the Peclet number (Pe = advection/diffusion), the Reynolds number (Re = inertial/viscous forces), the Fourier number (Fo = heat conduction rate), and the Biot number (Bi = internal/external thermal resistance). These help identify the dominant physical mechanisms and guide model simplification.

They are broadly categorized into Dirichlet (prescribed value, e.g., fixed temperature), Neumann (prescribed flux, e.g., heat flux or traction), and Robin (mixed or convective, e.g., heat transfer coefficient). Their mathematical treatment (essential vs. natural boundary conditions) affects the choice of numerical solution method (e.g., finite element vs. finite volume).