The main difference in implementing a linear viscous or any other kind of rate dependent model compared to a rate independent model arises in the evolution equation for stress and deformation resistance.

As we stated before the linear viscous model is a particular instance of the plastic potential function (5-4) with rate sensitivity m=l:

Ф = є0{-}2 (5-33)

s

In the linear viscous model, the plastic strain rate is:

ep =(lv)& (5-34)

where v = {,s/£0}is the viscosity in a linear viscous Maxwell body [37].

Then the relaxation equation yields:

& + {—)& = 0 (5-35)

3/1

Taking the trial effective stress a* as initial condition and integrating (5-34) over the time step we get:

(J = &* exp(--^) (5-36)

r

where:

r* =v/3ju (5-37)

is the relaxation time.

We estimate the macroscopic viscosity и for deformation of poly­crystals at high temperatures to be: kT, A G

The pre-exponential factor in (5-38) and the activation energy AG are data to be found experimentally. The most important data is AG that determines the order of magnitude of u. For plasticity above 0.8 Tm, where Tm is the melting point in Kelvin degree, it was found that for relatively coarse grain structures the activation energy for self diffusion is a good approximation for AG. This suggests that the effective mechanism of dislocation slip is diffusion controlled. Substituting default values for the rest of the unknowns does not affect the right order of magnitude of u. Reasonable values for those

defaults are the microscopic activation volume v* = 2|/?|3, Debeye frequency of thermal fluctuation vD=u (1/s), the density of mobile dislocations p = 1012 (1/m2), the average jump distance of a dislocation segment in one activation event (=b and the Burger’s vector length |Z>| = ал/2 / 2 for FCC and b = aV3 / 2 for BCC lattice, respectively, where a is the crystal lattice parameter.