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.