During deformation, the nucleation rate is evaluated as a function of both temperature T and strain rate. The growth of the new grains (mean grain size) is determined by both interface mobility and nucleation rate of small grains at the critical size. The F-A model uses the dislocation density model as input to estimate the driving force for the growth of the new recrystallized grains.

Nucleation and growth of recrystallization

The recrystallization model is based on dislocation density theory where the generated dislocations during deformation provide the driving force for recrystallization. The nucleation of the DRX occurs at grain boundaries once the accumulation of dislocations reaches the critical dislocation density defined by,

(1)

Where is high-angle grain boundary energy; is the dislocation mean free path, which is ~ for the most metals; is the dislocation line energy, where β is a constant of the order of 0.5; M_HB is the high-angle grain boundary mobility.

It is assumed that a nucleation event can occur during RX when a nucleus located on a grain boundary reaches a critical radius, R_c. The critical nucleus size corresponds to the condition when the stored energy difference between the DEF grain and the new RX grain or the nucleation driving force, , is large enough to overcome the capillary force of the nucleus. Then, the critical nucleus size is defined as,

(2)

The nucleation rate for DRX as a function of both temperature T and strain rate is calculated by [1993Pec]:

(3)

where C is the nucleation parameter,

(4)

with being the nucleation densities, D the grain size and N_f the nucleation site parameter, which can be estimated either by experiment or the inverse analysis method; Q_N is the activation energy of nucleation and exponent η is usually set to be 1. X_RX is the recrystallized volume fraction.

The nucleation rate for SRX (Static Recrystallization) is given by [2006Iva],

(5)

where is the deformation-stored energy and can be calculated from the dislocation density, ρ, as . E_c is the critical stored energy for initiating SRX, which can be determined from the critical deformation strain as , where the critical strain ε_c generally ranges from 0.05 to 0.1 for different materials.

Once nucleated, the growth rate of recrystallized grains, [m/s] is expressed by the product of an effective mobility for high angle grain boundary, M_eff, and the driving pressure as:

(6)

In Eq. 6, the effective grain boundary mobility linking grain boundary migration velocity to driving pressure is given by [2009Pay]:

(7)

Where D_eff is the effective self-diffusivity of the bulk atoms for a multi-component system, which is computed directly from CompuTherm mobility database; V_m is the molar volume. δ is the grain boundary width. A₂ is a factor related to the fraction of atomic jumps which result in a 'permanent' motion of the grain boundary in the direction of the driving force and a value of “1.0” is assumed. The driving pressure P contains one component P_C=–(2g_HB)/R related to the local grain boundary curvature. This pressure component is derived from the grain boundary energy and is also a function of the local grain boundary curvature. Additionally, the driving pressure P contains a term P_D=t[r] that is derived from the stored energy and hence related to the jump in dislocation density across the grain boundary. Finally, the driving pressure P will also be influenced by a term P_Z due to the presence of second phase precipitation. The influence of precipitates on recrystallization kinetics is incorporated via the Zener pressure, which is given for a spatial distribution of spherical precipitates by Neset al. [1985Nes] as with f the volume fraction of precipitates in mean radius r. Then, the total driving pressure can be calculated by,

(8)

In the Fast-Acting model, only the mean radius R and mean dislocation density ρ_DRX is tracked for the recrystallized grains. At each time step, the mean values must account concurrently for the increases of existing grains and the contributions due to the nucleation of the new grains.