Phase-field simulation of void evolution in UO<sub>2</sub> under applied stress

Jiang Yan-Bo; Liu Wen-Bo; Sun Zhi-Peng; La Yong-Xiao; Yun Di

doi:10.7498/aps.71.20211440

Abstract

Owing to the migration and aggregation of point defects produced by cascade collision, a large number of cavities form easily during irradiation of the uranium dioxide (UO₂) that is an important nuclear fuel. In addition, cavities are also inevitably introduced into the ceramic fuel during sintering. Moreover, the creep strain and thermal strain, caused by the extreme environment of high temperature and strong irradiation, significantly increase the applied stress of nuclear fuel. Therefore, it is crucial to investigate the microstructure evolution of the cavities in UO₂ fuel under applied stress. In this work, a phase-field model of void evolution in UO₂ under applied stress is established. Firstly, the elastic equilibrium equation is solved by the perturbation-iterative method, and the stress distribution around a single void under applied stress is calculated. The results show that the stress concentration is observed at the edge of the void, and the simulated stress distribution is consistent with the theoretically analytical results. Then, the evolution processes of a single void under different applied stresses are simulated by the phase-field model. The results show that the growth rate of void increases with the augment of applied stress. Finally, the effect of applied stress on grain growth and void evolution in polycrystalline are also studied. The results show that the applied stress will accelerate the void growth. With the increase of the applied stress, the effect of the applied stress on accelerating the void evolution increases.

Keywords:

Authors and contacts

1.
Institute of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China
2.
Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu 610213, China

Corresponding author: Liu Wen-Bo, liuwenbo@xjtu.edu.cn

Funds: Project supported by the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (Grant Nos. U1830124, U2130105) and the Innovative Scientific Program of China National Nuclear Corporation.

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References

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[8]	Brager H R, Garner F A, Guthrie G L 1977 J. Nucl. Mater. 66 301 Google Scholar
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[12]	Wiedersich H 1972 Radiat. Eff. Defect. S. 12 111 Google Scholar
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[42]	Aagesen L K, Gao Y, Schwen D, Ahmed K 2018 Phys. Rev. E 98 23309 Google Scholar
[43]	孙正阳, 王昱天, 柳文波 2020 金属学报 56 1643 Google Scholar Sun Z Y, Wang L T, Liu W B 2020 Acta Metall. Sin. 56 1643 Google Scholar
[44]	Liu W B, Ji Y Z, Tan P K, Zang H, He C H, Yun D, Zhang C, Yang Z G 2016 Materials 9 105 Google Scholar

Cited By

图 1 (a) 初始空洞示意图; (b) 空洞周围应力分布

Figure 1. (a) Schematic diagram of the initial void; (b) stress distribution with the void.

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图 2 三个典型截面上的应力分布

Figure 2. Stress distribution along three typical cross sections.

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图 3 模拟结果与解析解的应力分布比较

Figure 3. Comparison of stress distribution between simulation results and analytical solutions.

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图 4 (a)−(c)无应力下空洞演化; (d)−(f) 100 MPa附加应力下空洞演化; (g)−(i)应力分布随时间演化

Figure 4. (a)−(c) The evolution of void without stress; (d)−(f) the evolution of void with 100 MPa applied stress; (g)−(i) the distribution of stress during the evolution.

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图 5 不同应力下空洞半径随时间变化

Figure 5. The change of void radius with time under different stresses.

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图 6 (a)中线应力分布; (b)中线弹性能密度分布

Figure 6. (a) Stress distribution in the center line; (b) elastic energy density distribution in the center line.

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图 7 (a)初始结构示意图; (b)模拟区域的水平中线上的弹性模量分布

Figure 7. (a) Schematic of initial structure; (b) elastic constants distribution in the center line.

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图 8 (a)−(c)晶粒和空洞随时间演化; (d)−(f)演化过程中的应力分布

Figure 8. (a)−(c) Evolution of grains and void with time; (d)−(f) the distribution of stress during the evolution.

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图 9 晶间空洞演化中三个典型截面上应力分布

Figure 9. Stress distribution along three typical cross sections during intergranular void evolution.

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图 10 无外加应力下的双晶组织

Figure 10. The twin-crystal microstructure without applied stress.

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图 11 不同外加应力下的双晶组织((a)—(c))及对应的应力分布((d)—(f))　(a), (d) 50 MPa; (b), (e) 100 MPa; (c), (f) 150 MPa

Figure 11. The twin-crystal microstructure ((a)–(c)) and the corresponding stress distribution with different applied stresses ((d)–(f)): (a), (d) 50 MPa; (b), (e) 100 MPa; (c), (f) 150 MPa.

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图 12 无外加应力下的多晶组织

Figure 12. The polycrystal microstructure without applied stress.

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图 13 不同外加应力下的多晶组织((a)—(c))和对应的应力分布((d)—(f))　(a), (d) 50 MPa; (b), (e) 100 MPa; (c), (f) 150 MPa

Figure 13. The polycrystal microstructure ((a)–(c)) and the corresponding stress distribution with different applied stresses ((d)–(f)): (a), (d) 50 MPa; (b), (e) 100 MPa; (c), (f) 150 MPa.

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图 14 平均空洞半径随外加应力变化

Figure 14. The change of applied stress on average void radius

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表 1 模拟中使用的UO₂参数

Table 1. Parameters of UO₂ used in the simulation

参量	符号	取值	参考文献
绝对温度	T	1173 K
玻尔兹曼常数	k_B	1.381 × 10^–23 J/K
晶格常数	a	0.552 nm	[31]
迁移率	L	1.56 × 10^–11 m³/(J·s)	[32]
梯度自由能参数	κ	3.38 × 10^–8 J/m	[32]
空位扩散系数	$ {D_{\text{v}}} $	5.586 × 10^–18 m²/s	[33]
空位形成能	$ E_{\text{v}}^{\text{f}} $	5.10 eV	[34]
表面能	${\gamma _{\text{s}}}$	1.76 eV	[35]
晶界能	${\gamma _{{\text{GB}}}}$	1.67 eV	[36]
弹性模量	C₁₁	389.3 GPa	[37]
	C₁₂	118.7 GPa	[37]
	C₄₄	59.7 GPa	[37]

DownLoad: CSV

[1]	Govers K, Lemehov S, Hou M, Verwerft M 2007 J. Nucl. Mater. 366 161 Google Scholar
[2]	杨辉, 冯泽华, 王贺然, 张云鹏, 陈铮, 信天缘, 宋小蓉, 吴璐, 张静 2021 物理学报 70 054601 Google Scholar Yang H, Feng Z H, Wang H R, Zhang Y P, Chen Z, Xin T Y, Song X R, Wu L, Zhang J 2021 Acta Phys. Sin. 70 054601 Google Scholar
[3]	孙正阳, 杨超, 柳文波 2020 金属学报 56 1295 Sun Z Y, Yang C, Liu W B 2020 Acta Metall. Sin. 56 1295
[4]	Solomon A A 1973 J. Am. Ceram. Soc. 56 164 Google Scholar
[5]	Brask D N 1979 J. Nucl. Mater. 83 265 Google Scholar
[6]	Porter D L, Takata M L, Wood E L 1983 J. Nucl. Mater. 116 272 Google Scholar
[7]	Sahu H K, Jung P 1985 J. Nucl. Mater. 136 154 Google Scholar
[8]	Brager H R, Garner F A, Guthrie G L 1977 J. Nucl. Mater. 66 301 Google Scholar
[9]	Jiang Y, Liu W, Li W, Sun Z, Xin Y, Chen P, Yun D 2021 Comput. Mater. Sci. 188 110176 Google Scholar
[10]	Millett P C, Rokkam S, El-Azab A, Tonks M, Wolf D 2009 Model. Simul. Mater. Sci. Eng. 17 64003 Google Scholar
[11]	Brailsford A D, Bullough R, Hayns M R 1976 J. Nucl. Mater. 60 246 Google Scholar
[12]	Wiedersich H 1972 Radiat. Eff. Defect. S. 12 111 Google Scholar
[13]	Millett P C, El-Azab A, Rokkam S, Tonks M, Wolf D 2011 Comput. Mater. Sci. 50 949 Google Scholar
[14]	Millett P C, El-Azab A, Wolf D 2011 Comput. Mater. Sci. 50 960 Google Scholar
[15]	Liu W B, Wang N, Ji Y Z, Song P C, Zhang C, Yang Z G, Chen L Q 2016 J. Nucl. Mater. 479 316 Google Scholar
[16]	杨朝曦, 柳文波, 张璁雨, 贺新福, 孙正阳, 贾丽霞, 师田田, 恽迪 2021 物理学报 70 116101 Google Scholar Yang Z X, Liu W B, Zhang C Y, He X F, Sun Z Y, Jia L X, Shi T T, Yun D 2021 Acta Phys. Sin. 70 116101 Google Scholar
[17]	Hu S Y, Chen L Q 2001 Acta Mater. 49 1879 Google Scholar
[18]	Wang J J, Bhattacharyya S, Li Q, Heo T W, Ma X Q, Chen L 2012 Phil. Mag. Lett. 92 327
[19]	Kim D, Kim S G, Kim W T, Cho J, Han H N, Cha P 2011 Scr. Mater. 64 1079 Google Scholar
[20]	Chang K, Lee G, Kwon J 2016 Radiat. Eff. Defect. S. 171 242 Google Scholar
[21]	Salvo M, Sercombe J, Ménard J, Julien J, Helfer T, Désoyer T 2015 J. Nucl. Mater. 456 54 Google Scholar
[22]	Cahn J W, Hilliard J E 1958 J. Chem. Phys. 28 258 Google Scholar
[23]	Tonks M R, Zhang Y, Butterfield A, Bai X 2015 Model. Simul. Mater. Sci. Eng. 23 45009 Google Scholar
[24]	Wang Y U, Jin Y M, Khachaturyan A G 2002 J. Appl. Phys. 92 1351 Google Scholar
[25]	Sheng G, Bhattacharyya S, Zhang H, Chang K, Shang S L, Mathaudhu S N, Liu Z K, Chen L Q 2012 Mater. Sci. Eng. A 554 67 Google Scholar
[26]	Tonks M, Millett P 2011 Mater. Sci. Eng. A 528 4086 Google Scholar
[27]	Yan L L, Liu C Z, Ying Y Z, Zheng C 2014 Chinese Phys. B 23 69102 Google Scholar
[28]	Wang Y U, Jin Y M, Khachaturyan A G 2002 J. Appl. Phys. 91 6435 Google Scholar
[29]	Moelans N, Blanpain B, Wollants P 2008 Calphad 32 268 Google Scholar
[30]	Chen L Q, Shen J 1998 Comput. Phys. Commun. 108 147 Google Scholar
[31]	Kang K H, Ryu H J, Song K C, Yang M S 2002 J. Nucl. Mater. 301 242 Google Scholar
[32]	Aagesen L K, Schwen D, Tonks M R, Zhang Y 2019 Comput. Mater. Sci. 161 35 Google Scholar
[33]	Kinoshita M 1997 J. Nucl. Mater. 248 185 Google Scholar
[34]	Freyss M, Petit T, Crocombette J 2005 J. Nucl. Mater. 347 44 Google Scholar
[35]	Xiao H, Long C, Chen H, Tian X, Wei T, Zhao Y, Gao W 2015 Appl. Surf. Sci. 351 517 Google Scholar
[36]	Chen T, Chen D, Sencer B H, Shao L 2014 J. Nucl. Mater. 452 364 Google Scholar
[37]	Fritz I J 1976 J. Appl. Phys. 47 4353 Google Scholar
[38]	Jin Y M, Wang Y U, Khachaturyan A G 2003 Philos. Mag. 83 1587 Google Scholar
[39]	徐芝纶 1982 弹性力学(上卷) (北京: 高等教育出版社) 第89页 Xu Z L 1982 Elasticity (Vol. 1) (Beijing: Higher Education Press) p89 (in Chinese)
[40]	Wang H, Biswas S, Han Y, Tomar V 2018 Comput. Mater. Sci. 150 169 Google Scholar
[41]	Heo T W, Bhattacharyya S, Chen L 2013 Philos. Mag. 93 1468 Google Scholar
[42]	Aagesen L K, Gao Y, Schwen D, Ahmed K 2018 Phys. Rev. E 98 23309 Google Scholar
[43]	孙正阳, 王昱天, 柳文波 2020 金属学报 56 1643 Google Scholar Sun Z Y, Wang L T, Liu W B 2020 Acta Metall. Sin. 56 1643 Google Scholar
[44]	Liu W B, Ji Y Z, Tan P K, Zang H, He C H, Yun D, Zhang C, Yang Z G 2016 Materials 9 105 Google Scholar

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Phase-field simulation of void evolution in UO₂ under applied stress

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Corresponding author: Liu Wen-Bo, liuwenbo@xjtu.edu.cn

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Phase-field simulation of void evolution in UO2 under applied stress

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Corresponding author: Liu Wen-Bo, liuwenbo@xjtu.edu.cn

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Phase-field simulation of void evolution in UO₂ under applied stress