Generalized Theory for Diffusion-Induced Stress

The charging/discharging rate plays an important role in determining the structural integrity of metal-ion batteries, such as lithium-ion and sodium-ion batteries. In this work, we follow the approach by Green and Lindsay [Journal of Elasticity 2, 1 (1972)] and incorporate the contribution of the time derivative of the concentration of solute atoms to strain energy in Helmholtz and Gibbs free energies. Using the free energies, we obtain the chemical potential of the solute atoms and a generalized constitutive relation with the contribution of the concentration of the solute atoms and the time derivative of the concentration of the solute atoms. We use the generalized constitutive relation to analyze the stress evolution in an infinite, elastic cylinder under the condition that the stress-limited diffusion is negligible. The numerical results reveal that the term of the time derivative of the concentration of the solute atoms in the constitutive relation contributes to tensile hoop stress on the surface of the cylinder, which can initiate surface cracks and cause structural damage during the influx of the solute atoms. The generalized constitutive relation provides an approach to investigate the effect of charging/discharging rate on the structural integrity of electrodes in metal-ion batteries during electrochemical cycling.