Fig. 2: Effects of: (a) local normal stress σ n on
cavity growth rate; (b) cavity radius r andσ n on shrinkage rate at 500 ˚C (blue), 550 ˚C
(black) and 600 ˚C (red).
2.3 Numerical
implementation
Numerical computation of the cavity number density during creep-fatigue
was performed in a sectionalised manner. Let’s consider a time sectioni , the time period starts from (i -1)Δt and ends ati Δt , and a group of cavities can be nucleated during the
time interval Δt . This group of cavities are named as ‘cavities
of the i -th group’. All the cavities belonging to this group
would have the same radius and hence they are assigned as the nucleated
cavities within the time section i . The number density of
cavities of the i -th group is marked asρi , with its value equalling the product of
nucleation rate and Δt . The cavity is considered as nucleated
when its radius reaches the value of
2γ /σ n, and we assign the symbol0ri representing the critical
nucleation radius.
Since the growth rate is related to their radius, the cavities ofi -th group would have their characteristic rate at later time
sections. For a time section k (k >i ),
the cavity radius of the i -th group, symbolised ask -i ri , can be
worked out:
where is the growth rate at time section j(j >i ). This implies that cavities of thei -th group are considered as sintered, when the value ofk -i ri is less
than that of 0ri . Thus, the
sintered condition is defined as:
A flag variable (e.g. Flag(i ) for cavities of the i -th
group) is assigned to each cavity group as soon as cavities are
nucleated, and this will be updated at following time sections. The flag
value of 1 means that this group of cavities still exists, whilst 0
means that it is as sintered.
The number density of cavities ρ (k ) at the end of time
section k (k >i ) after taking sintering
into account can be formulised as:
where ρk is the number density of cavities
nucleated within the time section k . The cavitated GB area
fraction f (k ) at the end of time section k can be
then derived with the assumption of idealised hexagonal grain shape:
The coefficient of 106 has been generated due to the
unit conversion that involves the parameters of ρ in
mm-2, r in μm, and d in μm.