(1)

where is due to work hardening, is due to dynamic recovery. And the increase of the dislocation density during hot deformation is termed as,

(2)

Where f_w is the work hardening coefficient and relevant to the dislocation substructures, grain boundaries, precipitates, etc. ρ_i is the mean dislocation density for the DEF grain or the DRX grain. D is the average size of deformed grains and b is Burgers vector.

The dynamic recovery term is proportional to dislocation density,

(3)

where f_v is the dynamic recovery coefficient and depends on initial deformation conditions.

For the DRX grains, the evolution of dislocation density is also calculated by Eq. 1 with an additional softening factor f_x is introduced as [2020Cai],

(4)

Then the evolution of mean dislocation density can be obtained by,

(5)

where and are the volume fractions of DEF and DRX grains, respectively. Then, the relationship between the stress arising from the dislocation interactions, σ_M, and the mean dislocation density of the material, ρ_mean , can be described as,

(6)

where M is the Taylor factor. α is a constant depending on the dislocation/dislocation interaction and can be taken as 0.3. μ is shear modulus, and b is Burgers vector. Then the overall yield strength in Eq. 10 becomes .