# Toward Elucidating the Influence of Hydrostatic Pressure Dependent Swelling Behavior in the CERCER Composite

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Method

#### 2.1. Swelling Model with Recrystallization, Resolution and Hydrostatic Pressure

#### 2.2. Three-Dimensional Stress Update Algorithm

#### 2.3. Finite Element Model

#### 2.4. Simulations and Data Analyses

## 3. Results

#### 3.1. Model Verification

#### 3.2. The Swelling Behavior of Fission within the CERCER

#### 3.3. The Effects of Hydrostatic Pressure on the In-Pile Behavior of CERCER Composites

## 4. Discussion and Outlook

#### 4.1. The Swelling Behavior of Fission with the CERCER Composite

#### 4.2. Hydrostatic Pressure Effects on the In-Pile Behavior of the CERCER Composite

#### 4.3. Limitations and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Description of the Gas Swelling Model for $\mathit{U}{\mathit{O}}_{\mathbf{2}}$ Fuels

#### Appendix A.1. Governing Equations for Gas Diffusion

#### Appendix A.2. Calculation of Swelling for $U{O}_{2}$ Fuels

**Table A1.**The following parameters were used in the gas swelling model for swelling calculations, with recrystallization, resolution effect, and hydrostatic pressure dependency taken into account.

Parameter | Value | Unit |
---|---|---|

Y | 0.25 | - |

z | 4 | - |

a | $5.47\times {10}^{-10}$ | - |

$\lambda $ | $2.0\times {10}^{-8}$ | m |

$\gamma $ | 1 | N/m |

${D}_{0}$ | $1.0\times {10}^{-39}$ | m${}^{5}$ |

${r}_{g}$ | $2.16\times {10}^{-10}$ | m |

${b}_{v}$ | $8.5\times {10}^{-29}$ | m${}^{3}$/atom |

${h}_{s}$ | 0.6 | - |

${\delta}_{gb}$ | $2.0\times {10}^{-9}$ | m |

${r}_{gr0}$ | $7.5\times {10}^{-6}$ | m |

${r}_{grx}$ | $1.0\times {10}^{-5}$ | m |

${B}_{2}$ | $2.0\times {10}^{-34}$ | m${}^{5}$/N |

## Appendix B. Irradiation-Induced Creep in MgO Matrix

## References

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**Figure 1.**Finite element model of a CERCER composite pellet: (

**a**) diagram of a CERCER composite with a heterogeneous configuration of periodically distributed fuel particles; (

**b**) the RVE model; (

**c**) meshing geometry in ABAQUS; and (

**d**) grain-scale fission gas swelling with a spherical grain illustration showing the recrystallization process [14].

**Figure 2.**Temperature and swelling distribution contour plots on the 230th day of burnup: (

**a**) temperature without hydrostatic pressure; (

**b**) temperature with hydrostatic pressure; and (

**c**) swelling without hydrostatic pressure; (

**d**) swelling with hydrostatic pressure.

**Figure 3.**Model verification of cases with and without hydrostatic pressure: (

**a**) plots of volumetric swelling vs. hydrostatic pressure with FEM results (hollow dots) compared to theoretical calculations (solid lines); (

**b**) plots of volumetric swelling vs. burnup stages comparing UMAT results (dashed line) and theoretical calculations (hollow dots).

**Figure 4.**Investigations of multiscale fission swelling behavior in different simulation scenarios: (

**a**) fission gas swelling in the non-recrystallized and recrystallized regions; (

**b**) intergranular bubble radius in the non-recrystallized (${R}_{b}$) and recrystallized (${R}_{bx}$) regions.

**Figure 5.**Influences of the hydrostatic pressure: (

**a**) calculated hydrostatic pressure distribution along the particle path; (

**b**) temperature evolution; and (

**c**) fission gas swelling evolution.

**Figure 6.**Hydrostatic pressure influences: (

**a**) calculated the intergranular bubble radius in the non-recrystallized region as ${R}_{b}$ and (

**b**) intergranular bubble radius of ${R}_{bx}$ in recrystallized regions.

**Table 1.**For FE simulations, loading conditions such as the fission rate, fast neutron flux, and heat generation rate of fuel particles are used.

Fission Rate | Fast Neutron Flux | Heat Generation Rate |
---|---|---|

$2.5\times {10}^{20}\phantom{\rule{0.166667em}{0ex}}$ (fission/m${}^{3}$ s) | $2.5\times {10}^{15}\phantom{\rule{0.166667em}{0ex}}$ (n/cm${}^{2}$ s) | $8\phantom{\rule{0.166667em}{0ex}}$ (W/mm${}^{3}$) |

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**MDPI and ACS Style**

Zhao, J.; Chen, Z.; Zhao, Y.
Toward Elucidating the Influence of Hydrostatic Pressure Dependent Swelling Behavior in the CERCER Composite. *Materials* **2023**, *16*, 2644.
https://doi.org/10.3390/ma16072644

**AMA Style**

Zhao J, Chen Z, Zhao Y.
Toward Elucidating the Influence of Hydrostatic Pressure Dependent Swelling Behavior in the CERCER Composite. *Materials*. 2023; 16(7):2644.
https://doi.org/10.3390/ma16072644

**Chicago/Turabian Style**

Zhao, Jian, Zhenyue Chen, and Yunmei Zhao.
2023. "Toward Elucidating the Influence of Hydrostatic Pressure Dependent Swelling Behavior in the CERCER Composite" *Materials* 16, no. 7: 2644.
https://doi.org/10.3390/ma16072644