Rolling Simulation

It's an FEM analysis that predicts rolling force, torque, and thickness distribution in hot/cold rolling. The coupling between elastic deformation of the rolls (roll crown) and plastic deformation of the strip is extremely important.

Expressing this in a formula, it looks like this.

Contact arc length:

Rolling simulation is formulated as a coupled problem of thermodynamics, solid mechanics, and fluid mechanics. 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 in extrapolation ranges is verified. Thermodynamic databases like JMatPro and Thermo-Calc are also utilized.

Manufacturing process simulation is formulated as a coupled problem of thermodynamics, fluid mechanics, 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 in a formula, it looks like this.

Here, $L$ is the 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 like Johnson-Cook:

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