2.2.1 Method

The cavitated area fraction, w, along the grain boundaries is given as [1]:

$$
\pi w = \int \pi R^2 N(R, t) dR \tag{5}
$$

$$w = I(a, \beta, \gamma) A\_2 A\_1^{\frac{2}{\beta + 1}} t^{a + \beta + \frac{(1 - a)(\beta + \beta)}{\beta + 1}} \tag{6}$$

where the dimensionless factor Ið Þ α, β, γ is definite integral

$$I = \pi (\mathbf{1} + \boldsymbol{\beta})^{(\boldsymbol{\beta} + \mathbf{3})/(\boldsymbol{\beta} + 1)} \Big|\_{0}^{U} \mathbb{1}^{\boldsymbol{\beta} + 2} \Big[ \mathbf{1} - (\mathbf{1} - \boldsymbol{a}) \boldsymbol{\varkappa}^{\boldsymbol{\beta} + 1} \Big]^{(\boldsymbol{a} + \boldsymbol{\beta})/(\mathbf{1} - \boldsymbol{a})} d\boldsymbol{\varkappa} \tag{7}$$

and the cavity size distribution function, N Rð Þ , t , represents the number of voids with radii between R and þdR in the time interval t and t þ dt:

$$N(R,t) = \frac{A\_2}{A\_1} R^\beta t^{a+\gamma} \left(1 - \frac{1-a}{1+\beta} \frac{R^{\beta+1}}{A\_1 t^{1-a}}\right)^{(a+\gamma)/(1-a)}\tag{8}$$

if the non-station growth rate of the cavity radius and the nucleation rate of cavity are:

$$
\dot{R} = A\_1 R^{-\beta} t^{-a} \tag{9}
$$

$$J^\* = A\_2 t^\gamma \tag{10}$$

where the unknown constants A1, A2, α, β, and γ are material cavitation constants.

It is emphasized and concluded here that (1) if the values of creep cavitation constants A1, A2, α, β, and γ are known, the cavitated area fraction, w, could be determined quantitatively and (2) a critical value for wf <sup>¼</sup> <sup>π</sup> <sup>4</sup> for coalesce could be used as fracture criterion.

#### 2.2.2 Determination of cavitation constants

The method for the determination of the material cavitation constants, A1, A2, α, β, and γ, depends on the available experimental cavitation data [12].

The nucleation rate and growth rate can be directly determined if such data is directly available.

Based on qualitative analysis for 3D tomographic reconstructions of the distribution voids of E911 and P91 steel, a theoretically derived function of taking into account nucleation and growth of voids [1] was used to evaluate the experimental obtained histograms; the distribution equation (8) proposed by Riedel fitted well with the histogram density functions of void equivalent radius R of E911 and P91 [21], while the identical value of β = 1.95 � 0.05 (closely to 2) is characterized for the constrained diffusional mechanism of void growth and α = 1 characterized for continuum cavity nucleation [21, 22].

Cavity histogram is often used by material scientists, and it can be used to determine the values of these five or part of these constants, through either optimization method or trial and error method.
