# Modelling Irradiation Effects in Metallic Materials Using the Crystal Plasticity Theory—A Review

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## Abstract

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_{2}emissions. Then, it describes the impact of irradiation on the microstructure and mechanical properties of reactor structural materials. The main part provides the reader with a thorough overview of crystal plasticity models developed to address the irradiation effects so far. All three groups of the most important materials are included. Namely, the Zr alloys used for fuel cladding, austenitic stainless steels used for reactor internals, and ferritic steels used for reactor pressure vessels. Other materials, especially those considered for construction of future fission and fusion nuclear power plants, are also mentioned. The review also pays special attention to ion implantation and instrumented nanoindentation which are common ways to substitute costly and time-consuming neutron irradiation campaigns.

## 1. Introduction

_{2}emissions. The rate of this change seems to be too fast to be manageable by most of the flora and fauna species as well as modern human civilizations. Therefore, it is important to introduce emission-free energy sources [1]. Although solar and wind power plants are paid much attention to, they cannot play a dominant role yet as they are not sufficiently stable and large-scale storage of electrical energy is still an unsolved problem. Moreover, they require extremely large usage of land in order to fulfil country-scale energy needs and the rate of installing new power plants is not satisfactory. That is why at least some part of the energy should come from nuclear power plants (NPPs) if CO

_{2}emissions are to be reduced quickly. It should be also mentioned that apart from being emission-free, NPPs offer further advantages, such as an extremely low amount of fuel (as compared to fossil fuel power plants), which offers the possibility to store the fuel for many years of operation in advance, thus being independent from unstable political situations across the world; extremely low amounts of waste (as compared to fossil fuel power plants); stable electrical energy generation (also heat generation if cogeneration is considered); and extremely low usage of land (especially as compared to solar, wind and water power plants).

- Zirconium alloys having hexagonal close-packed (HCP) lattices. They are used for fuel cladding and thus are subjected to the highest radiation. On the other hand, they have to survive only during the time between subsequent fuel replacements (typically around 6 years).
- Austenitic stainless steels (ASS) having face-centred cubic (FCC) lattices. They are used for reactor vessel internals, which fulfil many functions such as supporting the core, control rod assemblies, core support structure, and reactor pressure vessel (RPV) surveilance capsules [3]. As they are inside the RPV, they are subjected to considerable neutron fluxes,
- Ferritic steels of body centred cubic (BCC) lattice, such as e.g., US A508C1 or A533B, French 16MnD5, Russian 15Cr2MoVA, and Chinese A508-3 steels, are used to build reactor pressure vessels. As the vessel is typically very large and has very thick walls, it is in principle the only part that cannot be replaced. Thus, its lifetime determines the service lifetime of the whole NPP.

- Longer operation times;
- Higher radiation doses;
- Higher operating temperatures (especially in the case of VHTR);
- More chemically aggressive environments.

## 2. Irradiation-Induced Effects

## 3. Modelling Irradiation Effects

#### 3.1. Materials for Fuel Cladding

#### 3.2. Model FCC Materials

#### 3.3. Materials for Reactor Internals

#### 3.4. Materials for Reactor Pressure Vessel

#### 3.5. Materials for Fusion

## 4. Nanoindentation

## 5. Conclusions

- Reproduce the experimentally observed irradiation hardening and post-yield softening;
- Develop analytical models of crack nucleation;
- Evaluate the influence of irradiation on DBTT, IGSCC as well as growth and coalescence of voids;
- Provide data for probabilistic assessment of brittle fracture.

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NPP | nuclear power plant |

PWR | pressurized water reactor |

HCP | hexagonal close-packed |

ASS | austenitic stainless steels |

FCC | face-centred cubic |

RPV | reactor pressure vessel |

BCC | body-centred cubic |

DBTT | ductile to brittle transition temperature |

Gen-IV | generation IV |

GIF | Generation-IV International Forum |

VHTR | very high temperature gas-cooled reactor |

GFR | gas-cooled fast reactor |

SFR | sodium-cooled fast reactor |

LFR | lead-cooled fast reactor |

MSR | molten salt reactor |

SCWR | super-critical water-cooled reactor |

FM | ferritic–martensitic |

ODS | oxide dispersion strengthened |

PKA | primary knock-on atom |

SFT | stacking fault tetrahedron |

DL | dislocation loop |

SRC | solute rich cluster |

DBH | dispersed barrier hardening |

dpa | displacement per atom |

PCP | phenomenological crystal plasticity |

DDCP | dislocation-density-based crystal plasticity |

CRSS | critical resolved shear stress |

RSS | resolved shear stress |

IGSCC | intergranular stress corrosion cracking |

IASCC | irradiation-assisted stress corrosion cracking |

DC | dislocation channel |

CPFEM | crystal plasticity finite element method |

FFT | fast Fourier transform |

EVPSC | elastic-viscoplastic self-consistent |

BZ | Berveiller and Zaoui |

SC | self-consistent |

SSD | statistically stored dislocation |

GND | geometrically necessary dislocation |

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**Figure 1.**Dislocation loops in irradiated pure iron visible in TEM micrograph [34].

**Figure 2.**(

**a**) Dislocation channels in irradiated and tension-deformed Zircaloy-2, (

**b**) dislocations and dislocation loops within the channel [39].

**Figure 3.**Engineering stress strain curves in iron and iron-chromium alloys in (

**a**) virgin and (

**b**) irradiated state [40]. Post-yield softening and ductility loss are clearly observed in the irradiated case.

**Figure 4.**He bubbles in FeCrNi alloy subjected to He implantation seen as bright regions in TEM micrograph [41].

**Figure 5.**Image showing the damage profile of tungsten implanted with He ions simulated with SRIM software superimposed on TEM image [102].

**Figure 6.**Illustration of the influence of irradiation on instrumented indentation data. Load-displacement curves: (

**a**) $\alpha $-Fe, (

**b**) CrFeV alloy [103].

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**MDPI and ACS Style**

Frydrych, K.
Modelling Irradiation Effects in Metallic Materials Using the Crystal Plasticity Theory—A Review. *Crystals* **2023**, *13*, 771.
https://doi.org/10.3390/cryst13050771

**AMA Style**

Frydrych K.
Modelling Irradiation Effects in Metallic Materials Using the Crystal Plasticity Theory—A Review. *Crystals*. 2023; 13(5):771.
https://doi.org/10.3390/cryst13050771

**Chicago/Turabian Style**

Frydrych, Karol.
2023. "Modelling Irradiation Effects in Metallic Materials Using the Crystal Plasticity Theory—A Review" *Crystals* 13, no. 5: 771.
https://doi.org/10.3390/cryst13050771