Solid-state batteries comprising a ceramic electrolyte and Li metal anode have the potential to deliver enhanced safety along with higher specific energies compared to liquid electrolyte Li-ion batteries. However, stiff, and strong ceramic electrolytes can suffer short circuits resulting from the penetration of Li filaments through the ceramic at charging currents above a critical current density. This is remarkable since the yield strength of Li is on the order of a few MPa while the ceramic electrolytes have strengths of many 100s of MPa and moduli in the GPa range. The failure of these Li-ion cells occurs via two interconnected processes: (i) formation of voids at the Li electrode/electrolyte interface and (ii) growth of Li filaments, that emanate from vicinity of these voids, into the electrolyte. Our work has involved developing coupled electrochemical-mechanical variational principles to understand how the electrochemistry of these cells drives mechanical failure. The focus is on developing an understanding of how well-established ideas such as Butler-Volmer kinetics need to be modified in the context of these solid-state batteries. The numerical solution of the variational principles provides insights into experimental observations, but numerous uncertainties remain with regards the microscale properties of the Li and solid electrolytes as well the mechanisms coupling mechanical deformations and electrochemistry.