PHASE-FIELD COHESIVE MODELING OF FRACTURE IN STORAGE PARTICLES OF LITHIUM-ION BATTERIES
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摘要: 锂电池充放电过程中, 锂离子的脱出和嵌入会引发电极颗粒的不均匀体积变化和机械应力. 上述锂离子扩散过程诱发的应力与电极颗粒的尺寸大小、截面形状和冲放电速率有关, 可能会导致电极颗粒出现裂缝起裂、扩展甚至断裂等力学失效, 对锂离子电池的容量和循环寿命等性能产生不利影响. 为准确模拟并预测电极颗粒的力学失效过程, 在笔者前期提出的统一相场理论框架内进一步考虑化学扩散、力学变形和裂缝演化等耦合过程, 建立化学–力学耦合相场内聚裂缝模型, 发展相应的多场有限元数值实现算法, 并应用于二维柱状和三维球体锂电池电极颗粒的力学失效分析. 由于同时涵括了基于强度的起裂准则、基于能量的扩展准则以及基于变分原理的裂缝路径判据, 这一模型不仅适用于带初始缺陷电极颗粒的开裂行为模拟, 而且适用于无初始缺陷电极颗粒的损伤破坏全过程分析. 数值计算结果表明, 相场内聚裂缝模型能够模拟锂离子扩散引发的电极颗粒裂缝起裂、扩展、汇聚等复杂演化过程, 可为锂离子电池电极颗粒的力学失效预测和优化设计提供有益的参考.Abstract: During charging and discharging of lithium-ion batteries (LIBs), lithium extraction and insertion induce inhomogeneous volume changes of storage particles, resulting in significant mechanical stresses. Dependent on the size and shape of storage particles as well as the recharging-charging rate, the diffusion-induced stress may lead to crack nucleation, propagation and even fracture of storage particles, yielding detrimental effects on the capacity and cycle life of LIBs. Aiming to simulate and predict the failure process of storage particles in LIBs, this work addresses a chemo-mechanically coupled phase-field cohesive zone model (PF-CZM) within the framework of the unified phase-field theory for damage and fracture. The numerical algorithm and computational implementation are also presented in the context of the multi-field finite element method, with applications to the modeling of mechanical failure of two-dimensional cylindrical and three-dimensional spherical storage particles in LIBs. As it intrinsically incorporates the strength-based nucleation criterion, the fracture energy-based propagation criterion and the variational principle based path chooser, the proposed PF-CZM applies not only to fracture analyses of pre-notched storage particles, but also to the simulation of the complete failure process of intact ones with no pre-defined defects. Extensive numerical results demonstrate that the proposed model is able to capture arbitrary crack configurations in storage particles due to evolution of Li-ion concentration, to predict the resulting mechanical failure of LIBs, and is useful for the optimal design of commercial LIBs.
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Key words:
- lithium-ion batteries /
- diffusion /
- chemo-mechanical failure /
- phase-field theory /
- cohesive zone model
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图 1 具有线性软化曲线的相场内聚裂缝模型
Figure 1. Phase-field cohesive zone model with a linear softening law
图 2 锂电池电极颗粒的扩散−开裂−变形耦合过程
Figure 2. Chemo-mechanical coupling in storage particles of LIBs
图 3 求解浓度–相场–位移耦合控制方程的交错迭代算法流程图
Figure 3. Flow chart of the staggered algorithm in solving the coupled governing equations
图 4 二维电极颗粒的几何及边界条件
Figure 4. A cylindrical storage particle: Geometry and boundary conditions
图 5 脱锂过程电极颗粒的应力状态
Figure 5. The mechanical stress state in the particle during Li extraction
图 6 电极颗粒弹性分析结果: 最大主应力随时间的变化曲线
Figure 6. Elastic results of the particle: Evolution of the maximum principal stress
图 7 电极颗粒的弹性分析结果: 静水压力(左)和平均浓度(右)等值线
Figure 7. Elastic analysis results of the storage particle: Isolines of the hydrostatic stress (left) and concentration (right)
图 8 含初始缺陷电极颗粒的裂缝不扩展模式: 裂缝相场和离子浓度云图(左:
$a_{0} = 0.1 $ μm; 右:$a_{0} = 1.0$ μm)Figure 8. Pre-notched particles with no cracking growth: Profiles of the crack phase-field and concentration field
图 9 含初始缺陷电极颗粒的裂缝不扩展模式: 最大主应力
Figure 9. Pre-notched particles with no cracking growth: Evolution of the maximum principal stress
图 10 含初始缺陷电极颗粒的裂缝不扩展模式: 最大浓度与最小浓度之差
$\Delta C_{\max}$ 以及充放电状态SOCFigure 10. Pre-notched particles with no cracking growth: Evolution of the maximum concentration difference and state of charging
图 11 含初始缺陷电极颗粒的裂缝不扩展模式: 离子浓度梯度
$-\nabla C$ 和静水应力梯度$\nabla \sigma_{_\text{H}}$ Figure 11. Pre-notched particles with no cracking growth: Gradients of the concentration and hydrostatic stress
图 12 含初始缺陷电极颗粒的单裂缝扩展模式:不同时刻的裂缝相场云图
Figure 12. Pre-notched storage particles with a single cracking growth: Profiles of the crack phase-field at various time instants
图 13 含初始缺陷电极颗粒的多裂缝扩展模式:不同时刻的裂缝相场云图
Figure 13. Pre-notched particles with multiple cracking growths: Profiles of the crack phase-field at various instants
图 14 含初始缺陷电极颗粒的多裂缝扩展模式: 裂缝尖端和电极表面的最大主应力随时间的变化曲线
Figure 14. Pre-notched particles with multiple cracking growths: Evolution of the principal stresses at the crack tip and surface
图 15 无初始缺陷电极颗粒的安全模式: 裂缝相场演化云图
Figure 15. Intact particles with no cracking grow: Profiles of the crack phase-field at various time instants
图 16 无初始缺陷电极颗粒的多裂缝模式: 裂缝相场演化云图
Figure 16. Intact particles with multiple cracking grows: Profiles of the crack phase-field at various time instants
图 17 无初始缺陷电极颗粒的多裂缝模式: 不同充放电速率下的最终裂缝相场云图
Figure 17. Intact particles with multiple cracking grows: Profiles of the crack phase-field for various charging rates
17 无初始缺陷电极颗粒的多裂缝模式: 不同充放电速率下的最终裂缝相场云图(续)
17. Intact particles with multiple cracking grows: Profiles of the crack phase-field for various charging rates.(continued)
图 18 截面面积相同、高宽比不同的电极颗粒: 最终裂缝相场云图
Figure 18. Damage profiles in storage particles with identical section area but of various aspect ratios
图 19 截面面积相同、高宽比不同的电极颗粒: 裂缝长度演化曲线
Figure 19. Crack lengths in storage particles of identical section area but of various aspect ratios
图 20 球体电极颗粒三维分析: 锂离子浓度分布示意图
Figure 20. Spherical storage particles: Concentration contour of Li-ion
图 21 球体电极颗粒三维分析: 弹性阶段的主应力云图
Figure 21. Spherical storage particles: Contours of principal stresses in the elastic stage
图 22 球体电极颗粒三维分析: 裂缝相场分布云图
Figure 22. Spherical storage particles: Profiles of the crack phase-field
图 23 球体电极颗粒三维分析:
$ X-Z$ 平面裂缝相场演化Figure 23. Spherical storage particles: Evolution of the crack phase-field on the
$ X-Z$ plane表 1 锰酸锂正极颗粒材料参数取值
Table 1. Material parameters of the storage particle.
Parameters Values Young's modulus $E_{0}$ 93 GPa Poisson's ratio $\nu_{0}$ 0.3 mass density $\rho$ 4140 kg/m3 maximum concentration $C_{\max}$ $2.29 \times 10^{4}$ mol/m3 partial molar volume $V_{_\text{H}}$ $3.497 \times 10^{-6}$ m3/mol diffusion coefficient $D$ $7.08 \times 10^{-15}$ m2/s temperature $T$ 300 K failure strength $f_{\text{t}}$ 1300 MPa fracture energy $G_{\text{f}}$ 10 N/m 
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表 2 不同时刻圆柱体电极颗粒的截面锂离子平均浓度
Table 2. Average concentration of lithium ions in the cylindrical storage particle at various time instants.
Time $t$/s Average concentration $\bar{C}$ 32 0.9911 365 0.8987 1000 0.7222
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