# A Crystal Plasticity Model with Irradiation Effect for the Mechanical Behavior of FCC Metals

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## Abstract

In this paper, a crystal plasticity model considering the irradiation effect based on the thermal activation theory is established. The evolutions of screw dislocations, edge dislocations, and stacking fault tetrahedrals (SFTs) (induced by irradiation) are included into the model. The interactions between dislocations and irradiation-induced SFTs are also considered. The constitutive model is numerically implemented on the ABAQUS platform through UMAT subroutine and applied to study the irradiation effect on the mechanical behavior of pure copper. The mechanical properties of single and polycrystalline copper are studied, and the simulation results show that the constitutive model can properly predict the mechanical behavior of irradiated pure copper. Especially for polycrystalline copper, the simulation results are in good agreement with the experimental data.

## Keywords

Crystal plasticity Irradiation effect Uniaxial tension Dislocation FCC crystal## 1 Introduction

In nuclear industry, the mechanical properties of metal materials are influenced by irradiation and temperature. After irradiation, metal materials with different crystal structures always show an irradiation effect, i.e., a significant increase in yield strength and a decrease in toughness [1, 2]. The irradiation effect becomes more pronounced as the irradiation dose increases. When the irradiation dose increases to a certain value, an upper yield point will appear and then the stress decreases. Moreover, the intensity of irradiation effect is also affected by the irradiation temperature. The results of the post-irradiation tensile testing exhibit stronger irradiation hardening and show that materials irradiated at room temperature have less ductility compared with the high temperature’s occasion [3, 4].

From the microperspective, materials in the irradiation environment are impacted by energetic particles, which incurs cascade reaction subsequently and results in a large number of nanoscale irradiation defects, such as vacancies, interstitial atoms, voids, precipitates, stacking fault tetrahedrals (SFTs), small dislocation loops, and so on [5]. The type and density of defect are related to the crystal structure. For example, Singh et al. [6] investigated the microstructure and associated tensile properties of irradiated FCC (Cu, Pd and 304 stainless steel) and BCC (Fe, Mo) metals by experiments, and revealed the differences and similarities of metals with different crystal structures after irradiation.

Otherwise, many studies show that the plastic deformation of the crystal is attributed to the dislocations slipping along the corresponding slip planes. Crystal plasticity theory is a basic theory for describing the plastic deformation at mesoscale, which connects the microscopic slip mechanism with the macroscopic plastic deformation behavior [7, 8, 9, 10]. The deformation mechanism of the irradiated FCC materials is greatly affected by the dislocations and irradiation defects. Cheong et al. [11] developed a dislocation-mechanics-based crystallographic theory to study the mechanical properties of thin polycrystalline Cu specimens. Though the irradiation effect has not been considered, the evolution model of screw and edge dislocations is useful for studying the mechanical behavior of FCC crystal. Arsenlis et al. [12] introduced an internal state variable model for the mechanical behavior of irradiated structural materials within a multi-scale framework. Using the model, the behavior of Cu tensile specimens with varying irradiation damage was simulated. De et al. [13] proposed a defect- and dislocation-density-based evolution model to capture the features of irradiation hardening as well as intra-granular softening. The Jacobian-free multi-scale method (JFMM) was further applied to improve the computational performance in the polycrystalline aggregate simulations by using a Newton–Krylov process. The mechanical responses of neutron-irradiated single and polycrystalline OFHC copper were studied and the model could capture experimentally observed grain-level phenomena. Chen et al. [14] investigated the mechanical behavior and crystallographic texture evolution of irradiated FCC metals based on a physical theoretical model. The study revealed the texture evolution along with different orientations before and after irradiation. It was concluded that irradiation-induced defects could affect the mechanical behavior and texture evolution of metals, both of which were closely related to irradiation hardening. Xiao et al. [15] established a tensorial crystal plasticity model with both irradiation and temperature effects. Using the model, the mechanical behavior and characteristics of irradiated polycrystals at different temperatures could be captured. Besides, Xiao et al. [16] also explored the effects of irradiation damage and crystal size on the mechanical behavior of FCC single crystals based on a unified size-dependent tensorial plasticity model. Krishna et al. [17] proposed a micromechanics-based model for copper subjected to neutron irradiation. The defect evolution was considered in the model to reflect the prevention and annihilation effects of defects on dislocations.

In this paper, a crystal plasticity model considering irradiation effect is established by tracking the evolutions of screw dislocations, edge dislocations, and SFTs induced by the irradiation. The model can also reflect the interactions between dislocations and irradiation defects and has been implemented numerically on the ABAQUS finite element platform. Using the model, the mechanical behaviors of single and polycrystalline copper before and after irradiation are studied and predicted.

## 2 Constitutive Model

### 2.1 Crystal Plasticity Theory

*N*is the number of all activated slip systems,

*is the second-order identical tensor, \(\gamma ^{\alpha }\) is the shear strain. \({{{\varvec{m}}}}^{\alpha }\) and \({{{\varvec{n}}}}^{\alpha }\) are the slip direction and normal-to-slip plane of the \(\alpha \) slip system, respectively. The slip direction and normal-to-slip plane after deformation are given by*

**I***p*and

*q*are flow-rule-related parameters,

*G*and \(G_0 \) represent the shear modulus at temperature

*T*and 0 K, respectively. Open image in new window and Open image in new window represent the lattice friction stress of each slip system at temperature

*T*and 0 K, respectively. It is assumed that Open image in new window is the same in each sliding system, so Open image in new window is simplified to Open image in new window . And the term \(g^{\alpha }\) represents the total slip resistance to dislocation motion, including the parts caused by dislocations and irradiation defects.

### 2.2 Evolution of Dislocations

### 2.3 Evolution of Defects

Irradiation defects in the FCC copper crystal are mainly SFTs [4, 5], with a low density of small dislocation loops. In this paper, the defect evolution model proposed by Krishna [17] is adopted where the SFT is regarded as the main irradiation defect. Besides, the annihilation of irradiation defects is due to the reactions between dislocations and SFTs. The full absorption of SFTs by screw dislocations and partial absorption of SFTs by edge dislocations are reported in many studies [19, 20, 21]. The absorption mechanism is simplified and then applied to the model. However, other mechanisms that lead to annihilation, such as localized plastic heating, or stress-assisted conversion from SFTs into Frank loops, have not been considered at present.

*dt*. So the annihilation area of the defects is

## 3 Finite Element Method

## 4 Material Parameters

There are 12 crystallographic equivalent slip systems {111} \(\langle 110\rangle \) in a FCC metal crystal. The dislocation density of typical fully annealed FCC pure metal is about \(2\times 10^{12}\ \hbox {m}^{-2}\) [17]. It is assumed that the initial densities of edge and screw dislocations in each slip system are the same, and the value is about \(80,000\hbox { mm}^{-2}\). The Open image in new window for unirradiated copper mainly depends on dislocations and is set as 20 MPa. For irradiated copper, besides the main radiation defect SFT, there are many other types of defects, such as voids, small clusters of self-interstitial atoms, Frank dislocation loops, precipitated phase, which would also impede the dislocation motion and result in irradiation hardening. The influence of SFT on slip resistance is considered by the evolution Eqs. (25)–(35). Besides, the influences of other defects on slip resistance also need to be considered. According to [11], the slip resistance caused by irradiation defects is assumed the same as the unirradiated lattice friction stress. Therefore, the Open image in new window for irradiated copper includes the influences of dislocations and irradiation defects, and its value is set as the sum of the 20 MPa caused by dislocations and the 20 MPa caused by irradiation effects, that is, 40 MPa.

Typical material parameters for Cu

Elastic modulus (MPa) | Flow parameters | Hardening parameters |
---|---|---|

\(C_{11} = 166{,}100\) | \({\dot{\gamma }}_0 = 10^{6}\hbox { s}^{-1}\) | \(\lambda = 0.3\) |

\(C_{\mathrm{1}2} = 121{,}900\) | \(Q_0 = 2.77\times 10^{-19}\hbox { J}\) | \(b^{\alpha }= 0.257\hbox { nm}\) |

\(C_{\mathrm{44}} = 75{,}600\) | \(\varphi = 0.7\) | \(\omega _1 = 1.5,\, \omega _2 = 1.2\) |

\(G_0 = 49{,}000\) | \(\omega _{\mathrm{i}1} = 1.6,\, \omega _{\mathrm{i}2} = 1.1\) | |

\(p = 0.2,\, q = 1.2\) | \(\rho _{\mathrm{e}}^0 = 80{,}000\hbox { mm}^{-2},\, \rho _{\mathrm{s}}^0 = 80{,}000\hbox { mm}^{-2}\) | |

\(d_{\mathrm{q}} = \hbox {2.4 nm}\) | \(C_{\mathrm{e}} = 0.5,\, K_{\mathrm{e}} = 0.014,\, d_{\mathrm{e}} = 1.0\hbox { nm}\) | |

\(d_{\mathrm{def}} = 2.5\hbox { nm}\) | \(C_{\mathrm{s}} = 0.5, \, K_{\mathrm{s}} = 0.028,\, d_{\mathrm{s}} = 5.0\hbox { nm}\) |

## 5 Simulation Results and Discussion

### 5.1 Mechanical Behavior of Single Crystal

From Figs. 3, 4 and 5, it is demonstrated that different orientations show different mechanical behaviors under uniaxial tensile simulation due to the anisotropy of single crystal. The highest yield strength occurs in [111]. It can also be concluded that the yield strength increases significantly in every orientation and grows with the increase of irradiation dose.

### 5.2 Mechanical Behavior of Polycrystal

The polycrystalline model is generated by Voroni principle, and the map of 60 grains is generated in the 0.45 mm \(\times \) 0.1 mm plane with an average 30-micron grain size. The average grain size is in accordance with the experimental sample.

The volume density and average size of SFTs in irradiated OFHC Cu are shown in Table 2. As can be seen, the volume density of SFTs increases accordingly with the increase of DPA. When DPA reaches a certain value, the volume density of SFTs is nearly saturated. Assuming that the initial density of defects in each system is the same and there are 12 slip systems in a FCC crystal, the \(N_{\mathrm{def}}^\alpha \) in each slip system is \( 2 \times 10^{22}\hbox { m}^{-3}\) when DPA is 0.01, and \(3.75\times 10^{22}\hbox { m}^{-3}\) when DPA is 0.1. The testing temperature is set to be 373 K, the shear modulus *G* decreases to 44,000 MPa and the temperature-dependent elastic constants \(C_{11}\) = 163,800 MPa, \(C_{12}\) = 120,400 MPa, \(C_{44 }\) = 73,500 MPa are calculated by Varshni’s formula [32]. Based on the experimental data for irradiated copper [1, 5], the coefficients \(h^{\alpha \beta }\) and \(i^{\upalpha \upbeta }\) are adjusted to reflect the influence of temperature on the evolution of dislocations and defects with \({\omega }_{1} = 1\) and \({\omega }_\mathrm{i1} = 1.1\). Then the uniaxial tensile mechanical behaviors of non-irradiated OFHC Cu as well as the OFHC Cu under the conditions of DPA = 0.01 and DPA = 0.1 are simulated respectively.

*k*.

Average size and volume density of SFTs under different values of DPA [5]

DPA | \(d_{\mathrm{def}} \hbox { (nm)}\) | \(N_{\mathrm{def}} \hbox { (10}^{23}\hbox { m}^{-3}\hbox {)}\) |
---|---|---|

0.01 | 2.3 | 2.4 |

0.1 | 2.4 | 4.5 |

0.2 | 2.6 | 4.5 |

0.3 | 2.4 | 4.3 |

## 6 Conclusion

- (1)
The constitutive model can reflect the anisotropy of single crystal and the mechanical behavior change of the irradiated material, such as the increment of yield strength.

- (2)
This constitutive model is able to reflect the effect of irradiation dose on the mechanical behavior of pure copper: with the increase of irradiation dose, the yield strength increases as well.

- (3)
For polycrystalline copper material, the simulation results are in good agreement with the experimental data.

## Notes

### Acknowledgements

The support of the National Natural Science Foundation of China (NSFC) under Grant No. 11202114, Beijing Higher Education Young Elite Teacher Project under Grant No. YETP0156 and Tsinghua University Initiative Scientific Research Program under Grant No. 2019Z08QCX06 are gratefully acknowledged.

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