Fig. A1 Spring-dashpot model 19
The local normal stress σ n is equivalent to the
stress concentration at GB particles, which gives:
where is the GB sliding rate. η p is damping
coefficient for the Norton power-law creep with the deformation rate as
formulised below:
where σ sa is the stress shared in the
neighbouring grains. An intermediate parameter, transient damping
coefficient , has been introduced to reduce the complexity when
calculating the . Then, at the next time t +Δt can be
calculated by σ s(t +Δt ) andσ n(t ) from previous time t :
For a sufficiently small time increment Δt , can be approximated
as a constant. Thus, the differential equation forσ n can be derived as:
Basically, Eq. is the solution of Eq., and that explicit form is more
convenient for numerical calculations.