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A cylindrical electrode is approximated as a long cylinder in most of existing models in which a generalized plane strain condition/plane strain is used. Based on the theory of elasticity, analytical expressions are derived for concentration distribution and stress component in a finite-length cylindrical electrode under galvanostatic operation. Using the superposition theorem, the Li-concentration is a sum of the concentration due to axial diffusion and the concentration due to lateral diffusion, and the separation of variable method is used to solve diffusion equations. By using the Boussinesq-Papkovich function, the generalized stress component distribution of a linearly combined product of the exponential-type Fourier-Bessel series is derived. The spatiotemporal distribution of concentration and diffusion-induced stresses are calculated in a cylindrical electrode with traction-free condition. The results are compared with the simulation results from a finite element software. For the concentration distribution, the numerical result and simulation result are almost the same. For the stress component, no significant difference exists between the two results, the largest relative difference for radial stress in the center is found to be about 4% and state of charge (SOC) = 17.9%. The radial stress decreases with radial position increasing, and decreases to zero at the surface, which is consistent with the results under the boundary condition. The hoop stress is tensile stress around the center of electrode, and becomes a compressive stress near the surface. Owing to the fact that the tensile hoop stress is attributed to the crack initiation, this implies that when plastic deformation is negligible, cracks first form in the center. The stress components with different length-to-radius ratios are calculated. It is found that the stress caused by lateral diffusion increases with length-to-radius ratio increasing, while the stress induced by axial diffusion decreases with length-to-radius ratio increasing. This is because the lateral diffusion has a greater influence on Li-concentration distribution in a cylinder electrode with length-to-radius ratio increasing.
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Keywords:
- lithium-ion batteries /
- finite-length cylindrical electrode /
- diffusion-induced stress /
- Boussinesq-Papkovich function
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图 1 有限柱形电极嵌锂示意图
Figure 1. Schematic of the finite-length cylindrical electrode structure.
图 2 不同充电状态下z = 0时锂离子浓度(a)和应力分量(b)—(d)沿径向分布曲线(实线为理论解; 圆圈图形为数值解)
Figure 2. Radial distribution of Li-concentration (a) and stress components (b)–(d) under different SOCs and z = 0 (solid lines, theoretical results; circle symbols, FEM results).
图 3 (a)不同充电状态下z = 0处von Mises应力分布; (b) z = 0, r = R处von Mises应力随无量纲时间的变化
Figure 3. (a) Radial distribution of normalized von Mises stress under different SOC and z = 0; (b) variation of normalized von Mises stress at z = 0, r = R with dimensionless time.
图 4 长径比为0.5 (a), 1 (b)柱形电极中z = 0无量纲应力分量径向分布曲线(实线, C1 ≠ 0, C2 ≠ 0; 圆形符号, C1 = 0; 方形符号, C2 = 0)
Figure 4. Radial distribution of normalized stress component for aspect ratios of L/R = 0.5 (a), 1 (b) (solid lines, C1 ≠ 0, C2 ≠ 0; circle symbols, C1 = 0; square symbols, C2 = 0).
图 5 不同长径比柱形电极中无量纲断裂能随时间的变化
Figure 5. Variation of strain energy normalized with fracture energy for the cylindrical electrodes with different aspect ratios of L/R.
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[1] Tarascon J M, Armand M 2001 Nature 414 359
Google Scholar
[2] Liu X H, Zhong L, Huang S, Mao S X, Zhu T, Huang J Y 2012 ACS Nano 6 1522
Google Scholar
[3] Maranchi J, Hepp A, Evans A, Nuhfer N, Kumta P 2006 J. Electrochem. Soc. 153 A1246
Google Scholar
[4] Ning G, Haran B, Popov B N 2003 J. Power Sources 117 160
Google Scholar
[5] Sun H, Xin G, Hu T, Yu M, Shao D, Sun X, Lian J 2014 Nat. Commun. 5 4526
Google Scholar
[6] Zhao Y, Stein P, Bai Y, et al. 2019 J. Power Sources 413 259
Google Scholar
[7] Jung S K, Hwang I, Chang D, Park K Y, Kim S J, Seong W M, Eum D, Park J, Kim B, Kim J 2020 Chem. Rev. 120 6684
Google Scholar
[8] Gan C, Zhang C, Liu P, Liu Y, Wen W, Liu B, Xie Q, Huang L, Luo X 2019 Electrochim. Acta 307 107
Google Scholar
[9] Jaramillo-Cabanzo D, Ajayi B, Meduri P, Sunkara M 2020 J. Phys. D: Appl. Phys. 54 083001
Google Scholar
[10] Xiao X, Liu P, Verbrugge M, Haftbaradaran H, Gao H 2011 J. Power Sources 196 1409
Google Scholar
[11] Xu Z L, Liu X M, Luo Y S, Zhou L M, Kim J K 2017 Prog. Mater. Sci. 90 1
Google Scholar
[12] Yang Y, Yuan W, Kang W, Ye Y, Pan Q, Zhang X, Ke Y, Wang C, Qiu Z, Tang Y 2020 Sustainable Energy Fuels 4 1577
Google Scholar
[13] Prussin S 1961 J. Appl. Phys. 32 1876
Google Scholar
[14] Li C M 1978 Metall Trans. A 9 1353
Google Scholar
[15] Deshpande R, Cheng Y T, Verbrugge M W 2010 J. Power Sources 195 5081
Google Scholar
[16] Song Y, Lu B, Ji X, Zhang J 2012 J. Electrochem. Soc. 159 A2060
Google Scholar
[17] Yang F 2005 Mater. Sci. Eng., A 409 153
Google Scholar
[18] Wang W, Lee S, Chen J 2002 J. Appl. Phys. 91 9584
Google Scholar
[19] Hao F, Fang D 2013 J. Appl. phys. 113 0130
Google Scholar
[20] Vanimisetti S K, Ramakrishnan N 2012 Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science 226 2192
Google Scholar
[21] Chen J, Wang H, Liew K, Shen S 2019 J. Appl. Mech. 86 041006
Google Scholar
[22] Li J, Lotfi N, Landers R G, Park J 2017 J. Electrochem. Soc. 164 A874
Google Scholar
[23] Planella F B, Ai W, Boyce A M, Ghosh A, Korotkin I, Sahu S, Sulzer V, Timms R, Tranter T G, Zyskin M, Cooper S J, Edge J S, Foster J M, Marinescu M, Wu N, Richardson G 2022 Prog. Energy 4 042003
Google Scholar
[24] Peng Y, Zhang K, Zheng B, Yang F. 2019 J. Energy Storage 25 100834
Google Scholar
[25] Crank J 1979 The Mathematics of Diffusion (New York: Oxford University Press) pp4, 5
[26] Sternberg E, Mcdowell E 1957 Q. Appl. Math. 14 381
Google Scholar
[27] Qi Y, Hector L G, James C, Kim K J 2014 J. Electrochem. Soc. 161 F3010
Google Scholar
[28] Zhang X, Shyy W, Sastry A M 2007 J. Electrochem. Soc. 154 A910
Google Scholar
[29] Cho J H, Picraux S T 2014 Nano lett. 14 3088
Google Scholar
[30] Kohanoff J, Galli G, Parrinello M 1991 J. Phys. IV 1 351
Google Scholar
[31] David W, Thackeray M, De Picciotto L, Goodenough J 1987 J. Solid State Chem. 67 316
Google Scholar
[32] Beaulieu L, Eberman K, Turner R, Krause L, Dahn J J E, Letters S S 2001 Electrochem. Solid-State Lett. 4 A137
Google Scholar
[33] Gu M, Yang H, Perea D E, Zhang J G, Zhang S, Wang C M 2014 Nano Lett. 14 4622
Google Scholar
[34] Ryu I, Choi J W, Cui Y, Nix W D 2011 J. Mech. Phys. Solids 59 1717
Google Scholar
[35] Liu X H, Zheng H, Zhong L, Huang S, Karki K, Zhang L Q, Liu Y, Kushima A, Liang W T, Wang J W, Cho J H, Epstein E, Dayeh S, Picraux. S. T, Zhu T, Li J, Sullivan J P, Cuming J, Wang C, Mao S X, Ye Z Z, Zhang S L, Tang Y 2011 Nano Lett. 11 3312
Google Scholar
[36] Wu Q L, Li J, Deshpande R D, Subramanian N, Rankin S E, Yang F, Cheng Y T 2012 J. Phys. Chem. C 116 18669
Google Scholar
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