A linearized model is developed for lithium ion batteries, relying on simplified characterizations of lithium transport in the electrolyte and through the interface between the electrolyte and the storage particles of the electrodes. The model is valid as a good approximation to the behavior of the battery when it operates near equilibrium, and can be used for both discharge and charging of the battery. The rate of extraction of lithium from and to the electrode storage particles can be estimated from the results of the model, information that can be used in turn to estimate the shrinkage and swelling stresses that develop in the particles. Given specified rates of extraction for spherical particles, maps of the resulting shrinkage and swelling stresses can be developed connecting their values to battery parameters such as particles size, diffusion coefficient, lithium partial molar volume, and particle elastic properties. Since a constant rate of extraction can only be achieved for a limited period of time until the concentration of lithium at the particle perimeter constrains the lithium mass transport, plots of the average state of charge in the particle versus time are also produced.
Purkayastha , R., & McMeeking, R. M. (2012). A linearized model for lithium ion batteries and maps for their performance and failure. Journal of Applied Mechanics , 79(3), [ 031021 ]. https://doi.org/10.1115/1.4005962