Most lithium-ion battery electrodes experience large volume changes caused by concentration changes within the host particles during charging and discharging. Electrode failure, in the form of fracture or decrepitation, can occur as a result of repeated volume changes. In this work, we first develop analytic solutions for the evolution of concentration and stresses within a spherical electrode element under charging-discharging conditions when the system thermodynamics are ideal (e.g., no repulsion forces are significant between intercalate species). Both interfacial (electrochemical) kinetics and intercalate diffusion are comprehended. Based on the analytic solutions, we propose tensile stress-based criteria for the initiation of cracks within a spherical insertion electrode. These criteria may help guide the development of new materials for lithium-ion batteries with enhanced mechanical durability and identify battery operating conditions that, when maintained, keep the mechanical stresses below acceptable values, thereby increasing cell life.