#### 3.1. Microstructure Evolution Model

The microstructure evolution model is a kind of empirical model to calculate the DRX fraction, dynamic recrystallized grain size, and average grain size during deformation, which is related to the strain, strain rate, and temperature. Therefore, an model is necessary to obtain the deformation and temperature distributions in the hot rolling process. In this study, single pass rolling models are established and rolling parameters are shown in

Table 1. The material of the plate is 7055 aluminum alloy, and its constitutive equation [

24] is shown in Equation (1). The initial rolling temperature is 410 °C and the inlet thickness is 250 mm. The heat transfer coefficient [

25] to the environment is 5 W·m

^{2}·K

^{−1} and the contact heat transfer coefficient to the work roll and emulsion is 30,000 W·m

^{2}·K

^{−1}.

where

$\dot{\epsilon}$ is strain rate,

σ is flow stress, and

T is temperature.

The DRX process begins when the strain of the material exceeds critical strain. In terms of micro perspective, the dislocation density is in relation to the strain and the DRX nuclei appear when reaching the critical dislocation density. Critical strain is connected with peak strain and they are both functions of temperature, strain, and strain rate in the models proposed by Sellars and Yada [

26,

27], as shown in Equations (2) and (3).

where

ε_{p} is peak strain,

ε_{c} is critical strain.

The DRX process mainly depends on two parameters: The DRX fraction

X_{d} and recrystallized grain size

D_{d}. In general, the value of the recrystallized grain size is much smaller than the initial grain size. Therefore, the initial grain can be intensively refined through sufficient DRX processing. The DRX fraction, recrystallized grain size, and average grain size can be calculated by Equations (4)–(7).

where

X_{d} is the DRX fraction,

ε_{0.5} is the strain corresponding to 50% DRX fraction,

D_{d} is the recrystallized grain size,

D_{0} is the initial grain size, and

$\overline{D}$ is the average grain size after deformation.

#### 3.2. Cellular Automata Model

During the hot rolling process, work hardening and dynamic recovery are common phenomena; DRX will take place when the dislocation density reaches the critical value, which has a significant role in grain refinement. In this study, probabilistic CA models are adopted to predict the variation of the microstructure and grain size during hot rolling. In the CA model, the complex object is divided into cells discrete in time and space and each cell is defined by its dislocation density, crystal orientation, temperature, and strain rate, in which cells with the same crystal orientation belong to a specific grain. The status values of each cell are updated at each CA time step according to the transition rules between the cell and its neighborhood. In the CA model, the temperature, strain, strain rate, and flow stress are derived from FEM results.

To acquire equiaxed grain and decrease microsegregation from casting, the homogenization treatment is conducted before hot rolling. The nucleation points are spread uniformly into the simulation zone and they grow into equiaxed grains in all directions with the same probability.

Figure 2 shows the initial microstructure for the grain size of 100 μm, which corresponds to a point at the macro level in a finite element model.

7055 aluminum alloy is a kind of material with high stacking fault energy, which plays a vital role in its softening mechanism in hot deformation. The softening mechanism during hot rolling consists of two processes: Dnamic recovery and DRX. The flow stress decreases in the macro level and the dislocation density reduces in the micro level. The stacking fault energy of the material significantly affects the variation of dislocation density and thus influences the softening mechanism. The dislocation density in a material provides the relationship between flow stress at the macro level and microstructure characteristics at the micro level, as shown in Equation (8). Therefore, the variation of dislocation density can be obtained by true stress-strain curves [

21]. A Kocks–Mecking (KM) model is used to describe the variation of dislocation density under different conditions [

28], as shown in Equation (9).

where

$\overline{\rho}$ is the average dislocation density,

ρ is the dislocation density,

k_{1} and

k_{2} are work hardening and dynamic softening coefficients, respectively;

α is the dislocation interaction term,

μ is the shear modulus, and

b is the modulus of Burger’s vector.

Newly dynamically recrystallized nuclei start to appear when the dislocation density exceeds critical value. The nucleation rate (

$\dot{n}$) can be calculated by models from Ding and Guo [

29,

30], as shown in Equation (10). The recrystallized nuclei start to grow into recrystallized grain after nucleation during the deformation process. The grain growth velocity (

v) [

31] is related to pressure (

p) and the grain boundary (GB) mobility (

M), as shown in Equation (11).

where

C is the material constant,

m is the sensitivity coefficient of strain rate, and

Q_{act} is active energy.