Lithium-ion (Li-ion henceforth) batteries application to high power and high capacity demanding systems still encounters limitations. This technology is based on the classical intercalation reaction, during which Li is inserted into or extracted from electrodes. During this process the hosting material experiences large volume changes, leading to mechanical degradation, loss of integrity and of electric contact with current collectors. Nano-structured and composite material electrodes may reduce such damaging phenomena.

Computational modeling and simulations may be helpful in designing new combinations of materials and geometries. To simulate a whole battery cell at the scale at which Lithium ions intercalate would be computationally unfeasible. A multi-scale approach is desirable and the Computational Homogenization (CH) technique is here adopted. This allows to track micro structural events that initiate damage and lead to macroscopic failure from macroscopic boundary conditions, while limiting the computational cost.

This contribution extends to large strains the framework of the CH developed in [1,2]. Temperature dependence is also included, towards modeling batteries under extreme conditions. The adopted approach originates from the fundamental balance laws (of mass, force, charge) and the weak formulation derived has a clear energy interpretation. Electroneutrality assumption has been taken into account. Maxwell's equations are considered in a quasi-static sense in a rigorous setting.

All the materials forming the multi-component porous electrode are clearly identified. Migration, diffusion, and intercalation of Lithium in the active particles are modeled. Constitutive assumptions, that emanate from a rigorous thermodynamic setting, complete the formulation.
After spatial discretization, a Backward Euler time-advancing algorithm has been implemented in FEM codes. Different finite elements have been designed to deal with the electrolyte, the electrodes and the reaction layer between them. Several case studies have been simulated to validate the implemented formulation.

References
[1] A. Salvadori, E. Bosco, and D. Grazioli. A computational homogenization approach for Li-ion battery
cells. Part 1 - Formulation. Journal of the Mechanics and Physics of Solids, 65:114-137, 2014.
[2] A. Salvadori, D. Grazioli, and M.G.D. Geers. Governing equations for a two-scale analysis of Li-ion
battery cells. International Journal of Solids and Structures, 59:90-109, 2015.