Irradiation-induced swelling stems from fission gas swelling and fission solid swelling.

Here, an empirical equation to calculate the fission solid swelling was used [6]:

$$\begin{array}{}{\displaystyle \frac{\mathit{\Delta}{V}_{solid}}{{V}_{0}}=4.0\times {10}^{-29}f}\end{array}$$(9)

where *f* in fission/m^{3} describes the fission density, which describes the nuclear fission number occurred in a unit volume of fuel meat.

The fission gas model follows Yi Cui et al. [12] considering the recrystallization, hydrostatic pressure dependence and intergranular gas atom re-solution in grain scale. The swelling model in this study is based on the Booth model [18], in which the fuel grain is regarded as a spherical one. Fission gas swelling includes two parts, i.e., intragranular swelling and intergranular swelling.

Before recrystallization, the fission gas swelling can be calculated by the following equation:

$$\begin{array}{}{\displaystyle \frac{\mathit{\Delta}{V}_{gas}}{V}={\left.\left(\frac{\mathit{\Delta}{V}_{\mathrm{int}ra}}{V}\right)\right|}_{{r}_{gr}}+{\left.\left(\frac{\mathit{\Delta}{V}_{\mathrm{int}er}}{V}\right)\right|}_{{r}_{gr}}}\end{array}$$(10)

where the intragranular swelling can be expressed as [19]:

$$\begin{array}{}{\displaystyle \left(\frac{\mathit{\Delta}{V}_{\mathrm{int}ra}}{{V}_{0}}\right)|\begin{array}{c}\\ {r}_{gr}\end{array}=\frac{4\pi}{3}{r}_{b}^{3}{c}_{b}}\end{array}$$(11)

where, *r*_{b} in m is the radius of the intragranular bubble; *c*_{b} in n/m^{3} is the fission gas bubble density, representing the number of fission gas bubbles in a unit volume of fuel meat. How to obtain the variables of *r*_{b} and *c*_{b} was presented in [12], which were calculated with the concentration of intergranular gas atoms.

The intergranular swelling takes the form as [20]:

$$\begin{array}{}{\displaystyle (\frac{\mathit{\Delta}{V}_{\mathrm{inter}}}{{V}_{0}})|\begin{array}{c}\\ {r}_{gr}\end{array}=\frac{2\pi {R}_{b}^{3}{C}_{b}}{{r}_{gr}}}\end{array}$$(12)

where, *C*_{b} in n/m^{2} is the intergranular bubble density, depicting the number of intergranular bubbles in a unit area of grain boundary; *R*_{b} in m is the radius of the intergranular bubble; *r*_{gr} in m is the grain radius.

Considering the influence of external hydrostatic pressure, the modified Van der Waals gas law is used [20]:

$$\begin{array}{}{\displaystyle \left(\frac{2\gamma}{{R}_{b}}+p\right)\left(\frac{4\pi {R}_{b}^{3}}{3}-{h}_{s}{b}_{v}{N}_{b}\right)={N}_{b}kT}\end{array}$$(13)

where, *γ* is the surface tension in N/m, *p* is the external hydrostatic pressure in Pa, *N*_{b} = *N*/*C*_{b} is the gas atom number per intergranular bubble and *N* depicts the concentration of intergranular fission gas atoms, *h*_{s} is the fitting parameter to make the Van der Waals equation equivalent to the hard-sphere equation of state [19], *b*_{v} is the Van der Waals constant for Xe. *N* is calculated with the analytical solution [12], which was obtained through solving the governing equations for diffusion of fission gas atoms in the equivalent spherical grain, where re-solution of the intergranular fission gas atoms is considered.

So,

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{R}_{b}}{\mathrm{\partial}p}=\frac{{h}_{s}{b}_{v}{N}_{b}{R}_{b}^{2}-\frac{4\pi}{3}{R}_{b}^{5}}{\frac{16\pi}{3}\gamma {R}_{b}^{3}+4\pi {R}_{b}^{4}p+2\gamma {h}_{s}{b}_{v}{N}_{b}}}\end{array}$$(14)

with which the hydrostatic pressure can be solved through nonlinear iterations, as stated in the next section.

Considering Eq. (13) is a nonlinear equation, for a certain magnitude of *p* the Newton’s method to get *R*_{b} should be performed.

After recrystallization, the original grain is divided into two parts including the recrystallized region and the unrecrystallized region. In the unrecrystallized region, the calculation of the gas swelling is similar to the calculation before recrystallization; while in the recrystallized region, it depends only on the intergranular swelling, because almost all fission gas atoms will transfer to the intergranular bubbles. The intergranular swelling can be expressed as [20]:

$$\begin{array}{}{\displaystyle \left(\frac{\mathit{\Delta}{V}_{\mathrm{int}er}}{{V}_{0}}\right)|\begin{array}{c}\\ {r}_{grx}\end{array}=\frac{4\pi {R}_{bx}^{3}}{3}\left(\frac{3{C}_{bx}}{2{r}_{grx}}+\frac{1}{8{r}_{grx}^{3}}\right)}\end{array}$$(15)

where, *r*_{grx} is the recrystallized grain radius, *R*_{bx} is the radius of the intergranular bubble in the recrystallized region; *C*_{bx} is the intergranular bubble density [12].

Similar to that for *R*_{b}, according to the modified Van der Waals gas law, $\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{R}_{bx}}{\mathrm{\partial}p}}\end{array}$ can be obtained as:

$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}{R}_{bx}}{\mathrm{\partial}p}=\frac{{h}_{s}{b}_{v}{N}_{bx}{R}_{bx}^{2}-\frac{4\pi}{3}{R}_{bx}^{5}}{\frac{16\pi}{3}\gamma {R}_{bx}^{3}+4\pi {R}_{bx}^{4}p+2\gamma {h}_{s}{b}_{v}{N}_{bx}}}\end{array}$$(16)

with which the hydrostatic pressure can be calculated through nonlinear iterations, as described in the next section.

Subsequently, the fission gas swelling can be calculated according to

$$\begin{array}{}{\displaystyle \frac{\mathit{\Delta}{V}_{gas}}{V}=(1-{V}_{r})\left[{\left.\left(\frac{\mathit{\Delta}{V}_{\mathrm{intra}}}{V}\right)\right|}_{{r}_{gr}}+{\left.\left(\frac{\mathit{\Delta}{V}_{\mathrm{int}er}}{V}\right)\right|}_{{r}_{gr}}\right]}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+{V}_{r}{\left.\left(\frac{\mathit{\Delta}{V}_{\mathrm{int}er}}{V}\right)\right|}_{{r}_{grx}}}\end{array}$$(17)

where, *V*_{r} is the volume fraction of the recrystallized region [21].

So, the volumetric swelling strain can be expressed as:

$$\begin{array}{}{\displaystyle \frac{\mathit{\Delta}V}{V}=\frac{\mathit{\Delta}{V}_{solid}}{V}+\frac{\mathit{\Delta}{V}_{gas}}{V}}\end{array}$$(18)

Then, the logarithmic volumetric swelling strain can be obtained as:

$$\begin{array}{}{\displaystyle {\theta}^{sw}=\mathrm{ln}(1+\frac{\mathit{\Delta}V}{V})}\end{array}$$(19)

Obviously, the logarithmic swelling strains are obtained as

$$\begin{array}{}{\displaystyle {\epsilon}_{ij}^{sw}=\frac{1}{3}{\delta}_{ij}{\theta}^{sw}}\end{array}$$(20)

which depicts that the irradiation swelling strain in all directions is the same.

Some of used parameters are listed in , while the other parameters can be found in Ref. [12].

Table 1 Parameters used in the simulation

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