2. Modelling approach

2.1 Theoretical modelling

Modelling cavitation under creep-fatigue contains two parts. The first part focuses on describing the relationship between the applied stress and local normal stress, and the cavity nucleation governed by vacancy accumulation. The second part concerns about the radius change of nucleated cavities during creep-fatigue loading. These two parts are integrated through a numerical framework to give the final prediction to the number density of cavities. The governing equations are described below.
The rate of cavity nucleation () on the particles at GB can be described as 14:
where Ω is the atomic volume and k is the Boltzmann constant. δD B is the GB thickness multiplied by its self-diffusion coefficient. γ is the free surface energy.F v is a shape factor related to the cavity volume, and it has a value of 0.1585 for cavity nucleation at the top of GB particles 14. ρ max is the maximum number density of potential nucleation sites, and its value can be worked out through the relation of 32, by assuming that grain has an idealised hexagonal shape. f bis the area fraction of GB particles, is their average radius (in μm), and d is the grain size (in μm). A coefficient of 106 appears due to the unit conversion when expressing the number density of cavities in mm-2.
The cavity nucleation rate in Eq. depends on three variables: number density of nucleated cavities, local normal stress and temperature, symbolised as ρ , σ n and T , respectively. Accordingly, the relationship between the local normal stress and applied shear stress is formulised as follow:
where , and are the applied shear stress, its change rate and the local normal stress at time t . The derivation of Eq. is described in Appendix A. is the local normal stress at time tt . Note Δt is the time interval chosen for the numerical computation that will be described in Section 2.3. τ is time constant defined as:
where ƞ b is the damping coefficient of GB sliding. k e is the elastic modulus, defined asG /[0.57d (1-v )]. is the transient creep damping coefficient that has a form ofƞ p/[σ s-f bσ n]n -1.G and v are shear modulus and Poisson’s ratio.ƞ p is the reciprocal of pre-exponential factor and n is the exponent in the Norton power-law creep equation. More details about the cavity nucleation model can be found in Hu, Xuan19.
To obtain the actual size of the nucleated cavities, the radius change rate needs to be determined during the subsequent loadings. For cavity growth, Chen 25 pointed out that GB sliding played an important role in the early-stage growth. The transient GB sliding activity could immediately wedge open the supercritical nuclei along the particles 33. According to this ‘crack sharpening’ mechanism, the cavity tip velocity is limited by surface diffusion3.
Fig. 1a shows a schematic diagram of the cavity nucleation and early-stage growth assisted by GB sliding with a rate of . Because of insufficient surface diffusion, the nucleated cavity tends to form an irregular crack-like shape. This implies that the growth rate is independent of the cavity radius, given that the process is limited by the vacancy diffusion rate near the cavity tip. Based on the work of Chuang, Kagawa 23, equations were formulised by Nix, Yu 34 to describe cavity growth rate controlled by the vacancy surface diffusion under low and high stresses:
where D S is the surface diffusion coefficient.δ S is the width of surface diffusion, which is related to atom volume Ω, throughδ S1/3. b is the limiting cavity radius, equalling the half of average cavity spacing, λ/2 23. λ is the average distance between two nucleation sites, and its value can be estimated as below for an idealised hexagonal grain shape: