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 t +Δt . 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 S=Ω1/3. 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: