Analytical analysis of concentration distribution and diffusion-induced stress of finite-length cylindrical electrode under galvanostatic operation

Abstract

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.