*2.2. Numerical Simulation*

DEFORM FE commercial software was used in simulating the 3D complex distributions of temperature and strain in both rod and plate. The roll for rolling process was considered as a rigid body, and the workpiece was assumed to be isotropic material. The basic equation governing the temperature distribution of a workpiece during hot rolling is represented as follows:

$$
\rho C\_p \frac{\partial T}{\partial t} = k \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + Q \tag{5}
$$

where ρ, *Cp*, *k*, and *Q* are the density, specific heat capacity, thermal conductivity of a workpiece, and volumetric rate of heat generation arising from the plastic deformation, respectively. Thermal properties of a workpiece such as thermal conductivity and specific heat were chosen from the library data provided by DEFORM FE software. That is, *k* and ρ*Cp* values were approximately 31 W·m<sup>−</sup>1K−<sup>1</sup> and 4.3 N·mm<sup>−</sup>2K−1, respectively. To solve the above governing equation, the boundary conditions for a workpiece are expressed as follows:

$$k\left(\frac{\partial T}{\partial t}\right)\_{surface} = \varepsilon \sigma (T^4 - T\_a^4) + h\_{conv}(T - T\_a) + h\_{cond}(T - T\_R) \tag{6}$$

where ε, σ, *Ta*, and *TR* are emissivity, Stefan–Boltzmann constant, ambient temperature, and roll temperature, respectively. Ambient and roll temperature is 25 ◦C, and ε is assumed to be 0.7. *hconv* and *hcond* are the convective heat transfer coefficient and conductive heat transfer coefficient, respectively, which is discussed in the next section. The shear friction coefficient of 0.6 was selected in the roll-workpiece interface [10,25,26], and other process parameters were identical to the experimental conditions, as given in Figure 4 and Table 1.

To shorten the calculation time, the quarter of full geometry was modeled stemming from the symmetrical condition of the rod and plate rolling processes, and a section of 500 mm in length was simulated. The caliber rolls had 400 mm in diameter, and the rolling speed was set as 10 RPM. The brick-type mesh was used, and the total number of mesh elements in the workpiece was approximately 21,600.
