Moreover, thermal activation is a stochastic process and is therefore formulated in terms of the activation free energy (which is based on the physical nature of the rearrangement), the thermal energy and the attempt frequency. A thorough discussion of the role and modeling of thermal activation in inelastic flow (in polycrystalline materials) has been given by Kocks, Argon, and Ashby (17).
where [[Omega].sub.o], the attempt frequency, is on the order of the Debye frequency; [Delta][F.sub.f] is the activation free energy for the transformations; k is the Boltzmann constant; and T is the absolute temperature.
Here [Mathematical Expression Omitted] is the viscoplastic strain rate; [Mathematical Expression Omitted] is the pre-exponential factor given by the product of the number density of shear transformation sites (D), the volume-averaged shear strain increment per transformation (2[[Gamma].sup.T][Omega]/V) and the attempt frequency ([[Omega].sub.o]); [Tau] is the applied shear stress and the other variables have been defined earlier.
where [[Omega].sub.o] is the attempt frequency and [Delta][F.sub.f] and [Delta][F.sub.b] are the activation energies for the forward and reverse transformations, respectively.